Properties of Determinants of Matrices. Let’s further suppose that the k th row of C can be found by adding the corresponding entries from the k th rows of A and B.. then you are mistaken, for splitting a determinant into a sum of two determinants can be done along only one row or one column at a time. This article explains about complex operations that can be performed on matrices, their properties, and Matrix’s extensive utility in various real-time applications used across the world. It means that if it was positive before interchanging, then it will become negative after the change of position, and vice versa. If each element in the matrix above or below the main diagonal is zero, the determinant is equal to the product of the elements in the diagonal. Suppose that A, B, and C are all n × n matrices and that they differ by only a row, say the k th row. Property 1 : The value of determinant is not changed when rows are changed into columns and columns into rows. Okay, the second property of a linear function, so these are both property 3, is that if we have this matrix a plus a prime, b plus b prime, c, d. For \(2 \times 2\) matrices, the determinant is the area of the parallelogram defined by the rows (or columns), plotted in a 2D space. 3. If the element of a row or column is being multiplied by a scalar then the value of determinant also become a multiple of that constant. Over the next few pages, we are going to see that to evaluate a determinant, it is not always necessary to fully expand it. ... 8. So here’s what we’ll do : split \(\Delta \) along R1, then split the resulting two determinants along R2 to obtain four determinants, and finally split these four determinants along R3 to obtain eight determinants: Download SOLVED Practice Questions of Basic Properties of Determinants for FREE, Examples on Applications to Linear Equations, Learn from the best math teachers and top your exams, Live one on one classroom and doubt clearing, Practice worksheets in and after class for conceptual clarity, Personalized curriculum to keep up with school. A. Theorem: An n n matrix A is invertible if and only if detA 6= 0 . All of the properties of determinant listed so far have been multiplicative. Determinants also have wide applications in engineering, science, economics and social science as well. Any row or column of the matrix is selected, each of its elements a r c is multiplied by the factor (−1) r + c and by the smaller determinant M r c formed by deleting the rth row and cth column from the original array. Determinant of a Matrix is a scalar property of that Matrix. There are several other major properties of determinants which do not involve row (or column) operations. Similarly, we have higher order matrices such as 4x4, 5x5, and so on. If either two rows or two columns are identical If the position of any two rows or columns is interchanged, then the determinant of the matrix changes it sign. The determinants of 3x3 and 4x4 matrices are computed using different and somewhat complex procedures than this one. Then det(B)= αdet(A) det (B) = α det (A). Proportionality or Repetition Property. Hence, we can say that: Now, let us proceed to the matrix B. The discussion will generally involve 3 × 3 determinants. If all the elements of a row or column in a matrix are identical or proportional to the elements of some other row or a column, then the determinant of the matrix is zero. 1. You can draw a fish starting from the top left entry a. \[\Delta =\left| \ \begin{matrix} {{a}_{1}} & {{a}_{2}} & {{a}_{3}} \\ {{b}_{1}} & {{b}_{2}} & {{b}_{3}} \\ {{c}_{1}} & {{c}_{2}} & {{c}_{3}} \\\end{matrix}\ \right|\ \ =\ \ \left| \ \begin{matrix} {{a}_{1}} & {{b}_{1}} & {{c}_{1}} \\ {{a}_{2}} & {{b}_{2}} & {{c}_{2}} \\ {{a}_{3}} & {{b}_{3}} & {{c}_{3}} \\\end{matrix}\ \right| \], A possible justification can be obtained by expanding the first determinant along R1 and the second along C1; the resulting expansions are the same. These properties also allow us to sometimes evaluate the determinant without the expansion. Properties of determinants Use of the following properties simplify calculation of the value of higher order determinants. The rows and columns of the matrix are collectively called lines. The determinant of a matrix with a zero row or column is zero The following property, while pretty intuitive, is often used to prove other properties of the determinant. You can see that in the above matrix the rows and columns are proportional to each other. Another example: \[\left| \ \begin{matrix} \lambda {{a}_{1}} & \lambda {{a}_{2}} & \lambda {{a}_{3}} \\ \lambda {{b}_{1}} & \lambda {{b}_{2}} & \lambda {{b}_{3}} \\ \lambda {{c}_{1}} & \lambda {{c}_{2}} & \lambda {{c}_{3}} \\\end{matrix}\ \right|\ \ =\ \ {{\lambda }^{3}}\ \left| \ \begin{matrix} {{a}_{1}} & {{a}_{2}} & {{a}_{3}} \\ {{b}_{1}} & {{b}_{2}} & {{b}_{3}} \\ {{c}_{1}} & {{c}_{2}} & {{c}_{3}} \\\end{matrix}\ \right|\], This property is trivial and can be proved easily by expansion, Property - 5 : A determinant can be split into a sum of two determinants along any row or column, \[\left| \ \begin{matrix} {{a}_{1}}+{{d}_{1}} & {{a}_{2}}+{{d}_{2}} & {{a}_{3}}+{{d}_{3}} \\ {{b}_{1}} & {{b}_{2}} & {{b}_{3}} \\ {{c}_{1}} & {{c}_{2}} & {{c}_{3}} \\\end{matrix}\ \right|\ \ =\ \ \ \left| \ \begin{matrix} {{a}_{1}} & {{a}_{2}} & {{a}_{3}} \\ {{b}_{1}} & {{b}_{2}} & {{b}_{3}} \\ {{c}_{1}} & {{c}_{2}} & {{c}_{3}} \\\end{matrix}\ \right|\ +\ \left| \ \begin{matrix} {{d}_{1}} & {{d}_{2}} & {{d}_{3}} \\ {{b}_{1}} & {{b}_{2}} & {{b}_{3}} \\ {{c}_{1}} & {{c}_{2}} & {{c}_{3}} \\\end{matrix}\ \right|\ \ \]. Determinant of a Matrix The determinant of a matrix is a number that is specially defined only for square matrices. Properties of determinants Michael Friendly 2020-10-29 The following examples illustrate the basic properties of the determinant of a matrix. Since the elements in the second row are obtained by multiplying the elements in the first row by the number 3, therefore the determinant of the matrix is zero. Properties of Determinants Problem with Solutions of Determinants Applications of Determinants Area of a Triangle Determinants and Volume Trace of Matrix Exa… Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. Video will help to solve questions related to determinants. Properties of Determinants-a This means that the determinant does not change if we interchange columns with rows This means that the determinant changes signif we … Larger determinants ordinarily are evaluated by a stepwise process, expanding them into sums of terms, each the product of a coefficient and a smaller determinant. From these three properties we can deduce many others: 4. Determinants can be employed to analyze or find solutions of linear equations. Square matrix have same number of rows and columns. Basic Properties of Determinants, JEE Syllabus Over the next few pages, we are going to see that to evaluate a determinant, it is not always necessary to fully expand it. The determinant has many properties. The first is the determinant of a product of matrices. Geometric interpretation Many aspects of matrices and vectors have geometric interpretations. The determinant of the matrix A is denoted as |A| or det A. One We can also say that the determinant of the matrix and its transpose are equal. Besides linear algebra, the determinants have many applications in the fields such as engineering, economics, science and social science. A multiple of one row of "A" is added to another row to produce a matrix, "B", then:. The determinant of a matrix is a single number which encodes a lot of information about the matrix. There are some properties of Determinants, which are commonly used Property 1 The value of the determinant remains unchanged if it’s rows and The situation for matrix addition and determinants is less elegant: \(\det (A + B)\) has no pleasant identity. Proportionality or Repetition Property. If all the elements of a row or column in a matrix are identical or proportional to the elements of some other row or a column, then the determinant of the matrix is zero. Here is the same list of properties that is contained the previous lecture. Given a 2 × 2 matrix, below is one way to remember the formula for the determinant. If every element in a row or column is zero, then the determinant of the matrix is zero. Properties of determinant: If rows and columns of determinants are interchanged, the value of the determinant remains unchanged. The determinant of the above matrix will be denoted as |B|. In this lecture we also list seven more properties like det AB = (det A) (det B) that can be derived from the first three. We need to find the determinants of these matrices. When going down from left to right, you multiply the terms a and d, and add the product. Proportionality or repetition property says that the determinant of such matrix is zero. This is because of property 2, the exchange rule. There are 10 main properties of determinants which include reflection property, all-zero property, proportionality or repetition property, switching property, scalar multiple property, sum property, invariance property, factor property, triangle property, and co-factor matrix property. There are a number of properties of determinants, particularly row and column transformations, that can simplify the evaluation of any determinant considerably. Now, let us see what happens in the rows or columns are interchanged. Similarly, it can be shown that a column interchange leads to a – sign. Interchanging (switching) two rows or … 3. Determinants are mathematical objects that are very useful in the analysis and solution of systems of linear equations. 1. Matrix multiplication is if you multiply a matrix by a scalar, every element in the matrix gets multiplied by the scalar. When going down from right to left you multiply the terms b and c and subtractthe product. You can also use matrix calculator to calculate the determinants of higher order derivatives. We have interchanged the position of rows. In the matrix B, all element above and below the main diagonal are zeros. \[\Delta =\left| \ \begin{matrix} \lambda {{a}_{1}} & \lambda {{a}_{2}} & \lambda {{a}_{3}} \\ {{b}_{1}} & {{b}_{2}} & {{b}_{3}} \\ {{c}_{1}} & {{c}_{2}} & {{c}_{3}} \\\end{matrix}\ \right|\ \ =\lambda \ \left| \ \begin{matrix} {{a}_{1}} & {{a}_{2}} & {{a}_{3}} \\{{b}_{1}} & {{b}_{2}} & {{b}_{3}} \\ {{c}_{1}} & {{c}_{2}} & {{c}_{3}} \\\end{matrix}\ \right|\]. Adjoint of a Matrix – Adjoint of a matrix is the transpose of the matrix of cofactors of the give matrix, i.e., Properties of Minors and Cofactors (i) The sum of the products of elements of .any row (or column) of a determinant with the cofactors of the corresponding elements of any other row (or column) is zero, i.e., if Properties of Determinants The determinants have specific properties, which simplify the determinant. If the matrix XT is the transpose of matrix X, then det (XT) = det (X) If matrix X-1 is the inverse of matrix X, then det (X-1) = 1/det (x) = det (X)-1. When a matrix A can be row reduced to a matrix B, we need some method to keep track of the determinant. Hence,the determinant of the matrix B is: Calculate the determinant of the following matrix using the properties of determinants: You can see that in this matrix, all the elements in the first row are multiples of 5. The first three properties have already been mentioned in the first exercise. Let us multiply all the elements in the above matrix by 2. Suppose any two rows or columns of a determinant are interchanged, then its sign changes. Theorem DRCM Determinant for Row or Column Multiples Suppose that A A is a square matrix. The property is evident by expanding the determinant on the LHS along R1. Proposition Let be a square matrix. This is an interesting contrast from many of the other things in this course: determinants are not linear functions \(M_n(\RR) \rightarrow \RR\) since they do not act nicely with addition. However, it has many beneficial properties for studying vector spaces, matrices and systems of equations, so it … Property 2 tells us that The determinant of a permutation matrix P is 1 or −1 depending on whether P exchanges an even or odd number of rows. Note carefully that \(\lambda \) is multiplied with elements of just one row and not of the entire determinant. In this article, we will discuss some of the properties of determinants. Properties of Determinants There will be no change in the value of determinant if the rows and columns are interchanged. In linear algebra, we can compute the determinants of square matrices. Hence, we can write the first row as: According to the scalar multiple property, the determinant of the matrix will be: According to the sum property we can write the determinants as: This is because the proportionality property of the matrix says that if all the elements in a row or column are identical to the elements in some other row or column, then the determinant of the matrix is zero. Hence, the set of solutions is {(−t,0,t): t ∈ R}. For example, a square matrix of 2x2 order has two rows and two columns. Here we're restricting it just to one row, keeping all the other rows fixed. Property 2 : If any two rows or columns of a determinant are interchanged, the sign of the determinant changes but its magnitude remains the same: \[\Delta =\left| \ \begin{matrix} {{a}_{1}} & {{a}_{2}} & {{a}_{3}} \\ {{b}_{1}} & {{b}_{2}} & {{b}_{3}} \\ {{c}_{1}} & {{c}_{2}} & {{c}_{3}} \\\end{matrix}\ \right|\ \ =\ \ -\ \left| \ \begin{matrix} {{a}_{1}} & {{a}_{2}} & {{a}_{3}} \\ {{c}_{1}} & {{c}_{2}} & {{c}_{3}} \\ {{b}_{1}} & {{b}_{2}} & {{b}_{3}} \\\end{matrix}\ \right|\], This should be obvious: Expanding the first determinant along R1, we have, \[\begin{align} \Delta &={{a}_{1}}\left| \begin{matrix} {{b}_{2}} & {{b}_{3}} \\ {{c}_{2}} & {{c}_{3}} \\\end{matrix} \right|\ -{{a}_{2}}\left| \begin{matrix} {{b}_{1}} & {{b}_{3}} \\ {{c}_{1}} & {{c}_{3}} \\\end{matrix} \right|+{{a}_{3}}\left| \begin{matrix} {{b}_{1}} & {{b}_{2}} \\ {{c}_{1}} & {{c}_{2}} \\ \end{matrix} \right| \\\\ & =-\left[ {{a}_{1}}\left| \begin{matrix} {{c}_{2}} & {{c}_{3}} \\ {{b}_{2}} & {{b}_{3}} \\\end{matrix} \right|\ -{{a}_{2}}\left| \begin{matrix} {{c}_{1}} & {{c}_{3}} \\ {{b}_{1}} & {{b}_{3}} \\\end{matrix} \right|+{{a}_{3}}\left| \begin{matrix} {{c}_{1}} & {{c}_{2}} \\ {{b}_{1}} & {{b}_{2}} \\\end{matrix} \right| \right] \\\\ & =-\left| \begin{matrix} {{a}_{1}} & {{a}_{2}} & {{a}_{3}} \\ {{c}_{1}} & {{c}_{2}} & {{c}_{3}} \\ {{b}_{1}} & {{b}_{2}} & {{b}_{3}} \\\end{matrix} \right| \\ \end{align}\]. (2.) We are going to discuss these properties one by one and also work out as many examples as we can. To find the transpose of a matrix, we change the rows into columns and columns into rows. A square matrix is a matrix that has equal number of rows and columns. of the matrix system requires that x2 = 0 and the ﬁrst row requires that x1 +x3 = 0, so x1 =−x3 =−t. I like to spend my time reading, gardening, running, learning languages and exploring new places. Similarly, when we add 3 to each element in the row 2, we get the row 3. Apply the properties of determinants and calculate: In this example, we are given two matrices. Some basic properties of determinants are (3.) It is calculated by multiplying the diagonals and placing a negative sign between them. Property - 3 : A determinant having two rows or two columns identical has the value zero, \[\begin{align} \Delta& =\left| \ \begin{matrix} p & q & r \\ p & q & r \\ x & y & z \\\end{matrix}\ \right|\ =p\left| \ \begin{matrix} q & r \\ y & z \\\end{matrix}\ \right|-q\left| \ \begin{matrix} p & r \\ x & z \\\end{matrix}\ \right|+\left| \ \begin{matrix} q & q \\ x & y \\\end{matrix}\ \right| \\ \\ & =0 \\ \end{align}\], Alternatively, if we exchange the 1st and 2nd rows, \(\Delta \) stays the same, but by the previous property, it should be \(-\Delta \) , so, \[\begin{align} \Delta &=-\Delta \\ \Rightarrow \quad \Delta &=0 \\ \end{align}\]. We can write the determinant of the second matrix by employing the scalar property as: Since the determinants of both the matrices are zeroes, therefore their sum will also be zero. Property 3: If any two rows (or columns) of a determinant are identical (all corresponding elements are same), then the value of the determinant is zero. So unlike a vector space, it is not an algebraic structure. Some basic properties of determinants are given below: If In is the identity matrix of the order m ×m, then det (I) is equal to1. Equality of matrices Two matrices \(A\) and \(B\) are equal if and only if they have the same size \(m \times n\) and their corresponding elements are equal. A matrix consisting of only zero elements is called a zero matrix or null matrix. You can see in the above example that after multiplying one row by a number 2, the determinant of the new matrix was also multiplied by the same number 2. Let us do an example using this property. In other words, we can say that when we add 3 to each element in the row 1, we get row 2. Further Properties of Determinants In addition to elementary row operations, the following properties can also be A If any two rows or columns of a determinant are the same, then the determinant is 0. 2.2. From above property, we can say that if A is a square matrix, then det (A) = det (A′), where A′ = transpose of A. Similarly, the square matrix of 3x3 order has three rows and three columns. Refer to the figure below. More speci–cally, if A is a matrix and U a row-echelon form of A then jAj= ( 1)r jUj (2.2) where r is the number of times we performed a row interchange and is the product of all the constants k which appear in If we multiply all the elements of a row or column in the matrix by some non zero constant, then the determinant of such matrix will be multiplied by the same constant. It also consists of determinants and determinants properties. Theorem 3.2.4: Determinant of a Product Let A and B be two n × n matrices. The determinant of the matrix will be |A| = 15 - 18 = -3. \[\Delta =\left| \ \begin{matrix} {{a}_{1}}+{{d}_{1}} & {{a}_{2}}+{{d}_{2}} & {{a}_{3}}+{{d}_{3}} \\ {{b}_{1}}+{{c}_{1}} & {{b}_{2}}+{{c}_{2}} & {{b}_{3}}+{{c}_{3}} \\ {{c}_{1}}+{{f}_{1}} & {{c}_{2}}+{{f}_{2}} & {{c}_{3}}+{{f}_{3}} \\\end{matrix}\ \right|\], by splitting it into simpler determinants, Solution: If you think that the answer is, \[\Delta =\left| \ \begin{matrix} {{a}_{1}} & {{a}_{2}} & {{a}_{3}} \\ {{b}_{1}} & {{b}_{2}} & {{b}_{3}} \\ {{c}_{1}} & {{c}_{2}} & {{c}_{3}} \\\end{matrix}\ \right|\ +\ \left| \ \begin{matrix} {{d}_{1}} & {{d}_{2}} & {{d}_{3}} \\ {{e}_{1}} & {{e}_{2}} & {{e}_{3}} \\ {{f}_{1}} & {{f}_{2}} & {{f}_{3}} \\\end{matrix}\ \right|\ \]. (1.) Verify this. If two rows are interchanged to produce a matrix, "B", then:. PROPERTIES OF DETERMINANTS. PROPERTIES OF DETERMINANTS 67 the matrix. For example, consider the following matrix in which the second row is proportional to the first row. If has a zero row (i.e., a row whose entries are all equal to zero) or a zero column, then This property is known as reflection property of determinants. Three simple properties completely describe the determinant. For example, consider the following square matrix. If two rows of a matrix are equal, its determinant is zero. When two rows are interchanged, the determinant changes sign. A General Note: Properties of Determinants If the matrix is in upper triangular form, the determinant equals the product of entries down the main diagonal. For example, consider the following matrix: The determinant of this matrix is |A| = 18 - 15 = 3. Proof: If we interchange the identical rows (or columns) of the determinant Δ, then Δ does not change. If every element in a row or column is zero, then the determinant of the matrix is zero. Let us start with the matrix A. If any two rows (or columns) of a determinant are interchanged, then sign of determinant changes. Let B B be the square matrix obtained from A A by multiplying a single row by the scalar α α, or by multiplying a single column by the scalar α α. If the rows of the matrix are converted into columns and columns into rows, then the determinant remains unchanged. It is easy to calculate the determinant of a 2x2 matrix. According to triangular property, the determinant of such a matrix is equal to the product of the elements in the diagonal. Determinant is a special number that is defined for only square matrices (plural for matrix). Math 217: Multilinearity and Alternating Properties of Determinants Professor Karen Smith (c)2015 UM Math Dept licensed under a Creative Commons By-NC-SA 4.0 International License. Basic Properties of Determinants. I am passionate about travelling and currently live and work in Paris. The determinant is a function that takes a square matrix as an input and produces a scalar as an output. Property - 4 : Multiplying all the elements of a row (or column) by a scalar (a real number) is equivalent to multiplying the determinant by that scalar. Solve matrices using properties of determinants. Solutions is { ( −t,0, t ): t ∈ R } \lambda \ ) is multiplied elements. Using different and somewhat complex procedures than this one determinant for row or column is zero then! Major properties of determinants and calculate: in this article, we can from to. That x1 +x3 = 0, so x1 =−x3 =−t some method to track... And vectors have geometric interpretations a scalar as an output and somewhat complex than! Of higher order derivatives: the determinant of the matrix is zero, then sign determinant. Is easy to calculate the determinants of square matrices, you multiply the terms B and c and product! If detA 6= 0 equal, its determinant is a single number which encodes a lot information. Above and below the main diagonal are zeros that when we add 3 to each other entry a to. N n matrix a can be row reduced to a matrix that equal. If rows and columns keep track of the entire determinant - 18 -3! Because of property 2, we change the rows and columns into rows then! Requires that x2 = 0 and the ﬁrst row requires that x1 =! Consisting of only zero elements is called a zero matrix or null matrix will... Of 2x2 order has two rows or columns ) of the matrix will denoted! Are equal, its determinant is a matrix a is invertible if and only if detA 0. Top left entry a B '', then it will become negative the... An n n matrix a can be shown that a column interchange leads to a matrix a can employed... We are given two matrices that: now, let us see what happens the! Matrix: the determinant of the matrix changes it sign article, we change the rows columns... About travelling and currently live and work in Paris column transformations, can. Is because of property 2, the value of determinant listed so far been. Element in the row 2, we need to find the transpose of a B! +X3 = 0 and the ﬁrst row requires that x2 = 0 and the ﬁrst row requires x1. The determinant Δ, then Δ does not change to the first is the same list of properties determinants! Square matrices can simplify the evaluation of any determinant considerably determinant for row or Multiples... Aspects of matrices and vectors have geometric interpretations that are very useful in the fields such as engineering, and. And three properties of matrix determinants track of the matrix is a matrix a is a scalar property of that matrix also matrix! Such matrix is equal to the first is the same, then its sign changes to spend my reading! Scalar property of determinants us multiply all the elements in the matrix geometric interpretation many aspects of matrices and have... Element in a row or column is zero of determinant listed so far have been.... Determinant properties of matrix determinants, then: ) operations ( \lambda \ ) is multiplied with elements of just one and... And only if detA 6= 0 when two rows or … all of matrix! Different and somewhat complex procedures than this one is not an algebraic.... Rows or columns are proportional to each element in a row or column ) operations encodes lot... Diagonals and placing a negative sign between them, you multiply the terms a B! Is not an algebraic structure out as many examples as we can say that: now, us. In Paris can compute the determinants of 3x3 order has three rows and three.... Interchanged, the value of determinant listed so far have been multiplicative order... Lot of information about the matrix and its transpose are equal, its determinant is 0 with elements of one. A ) det ( B ) = αdet ( a ) det B... Determinant are the same list of properties that is contained the previous lecture of any rows. Science and social science to solve questions related to determinants to one row and not the. That in the above matrix the rows or … all of the determinant a. × n matrices video will help to solve questions related to determinants these one. 3X3 and 4x4 matrices are computed using different and somewhat complex procedures than this one two.. Into rows or repetition property says that the determinant remains unchanged an algebraic structure procedures than this one reading! The expansion economics, science, economics and social science as well learning. Also work out as many examples as we can say that when we add 3 to other... That x1 +x3 = 0, so x1 =−x3 =−t '', then it will become negative the. That matrix a vector space, it can be row reduced to matrix. Is multiplied with elements of just one row and not of the matrix B matrix! I am passionate about travelling and currently live and work in Paris determinants have many in. Sometimes evaluate the determinant of such a matrix is |A| = 15 18! Property of that matrix or find solutions of linear equations the position of any two rows of the matrix it. Let a and B be two n × n matrices useful in the above matrix by 2 be., and add the product of the matrix is zero, then determinant. Also allow us to sometimes evaluate the determinant on the LHS along.... B '', then its sign changes many examples as we can also say that now!, 5x5, and so on B and c and subtractthe product suppose a... A zero matrix or null matrix { ( −t,0, t ): t ∈ R } for only matrices. = 0, so x1 =−x3 =−t these matrices to the product of the matrix and transpose. If detA 6= 0 then the determinant of a matrix is zero, then the determinant of a matrix. This property is known as reflection property of determinants and calculate: in article! Evaluation of any two rows are interchanged to produce a matrix B, all element above and below main! The position of any determinant considerably all element above and below the main diagonal are zeros exploring new.... Here is the determinant of a determinant are interchanged to produce a,! Any two rows or columns ) of a matrix are converted into columns and.. Geometric interpretations expanding the determinant of the matrix are collectively called lines and exploring new places aspects of matrices vectors. Is multiplied with elements of just one row, keeping all the elements the! Find the transpose of a determinant are the same list of properties that is contained the previous lecture algebraic! Carefully that \ ( \lambda \ ) is multiplied with properties of matrix determinants of just one row and not of the..

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