Chapter 2: Determinants 17

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1 Chapter 2: Determinants 17 SECTION B Properties of a Determinant By the end of this section you will be able to prove that the determinant of a triangular or diagonal matrix is the product of the leading diagonal entries evaluate determinants of a triangular and diagonal matrix prove properties of determinant of an elementary matrix establish certain properties of determinants of other matrices You will need to remember the definition of a determinant of a matrix and the technique to evaluate it by using cofactors. This section is a difficult section to follow because we need to prove a number of propositions and some of these rely on the results of chapter 1. B1 Revision of Properties of a Determinant In the last section we established certain properties of the determinant of a matrix such as: Proposition (2.10). If a square n by n matrix A consists of two identical rows then det = 0. Further properties were also established in the associated Exercise 2(a) and we give these reference numbers. T Proposition (2.14). Let A be a square matrix then det ( A ) = det ( A ). What does this mean? The determinant of the transposed matrix is the same as the determinant of the initial matrix. Proposition (2.15). Let B be a matrix obtained from matrix A by multiplying one row (or column) of A by a non-zero scalar k then det ( B) = k det ( A ) We will restate this result later in this section. Proposition (2.16). Let A be a square n by n matrix and k be a scalar then n det ( ka) = k det ( A ) What does this mean? The determinant of the scalar multiplication ka is times the determinant of the matrix A. T Note that since det ( A ) = det ( A ), expanding along a row or column is equivalent. Generally propositions about the determinant of the matrix with the word row can be T det A = det A. swapped by the word column because B2 Determinant Properties of Particular Matrices We can find determinants of particular matrices such as triangular matrices. What are triangular matrices? Definition (2.17). A triangular matrix is a n by n matrix where all entries to one side of the leading diagonal are zero. What is meant by the leading diagonal? Leading diagonal of a matrix is the entries going from the top left hand corner to the bottom right hand corner of the matrix, For example, the following are triangular matrices: n k

2 Chapter 2: Determinants Leading (a) and (b) Diagonal (a) is an example of an upper triangular matrix. (b) is an example of a lower triangular matrix. Another type of matrix is a diagonal matrix. Do you know what is meant by a diagonal matrix? Definition (2.18). A diagonal matrix is a n by n matrix where all entries to both sides of the leading diagonal are zero. Can you think of an example of a diagonal matrix? The identity matrix = Another example is I A diagonal matrix is both an upper and lower triangular matrix. Triangular and diagonal matrices have the following property. Proposition (2.19). The determinant of a triangular or diagonal matrix is a product of the entries along the leading diagonal. What does this proposition mean? Let A be an upper triangle matrix such that a11 a12 a1 n 0 a22 a2n A = 0 0 ann then det = a11a22a33 ann. We prove it for the upper triangular matrix and the proof for the lower triangular matrix is similar. Since the diagonal matrix is a particular upper (or lower) triangular matrix therefore the proof of the diagonal matrix follows from the upper triangular result. We use proof by induction. Remember the method of proof by induction is threefold. 1. Prove it for a base n= 1 or n= 2 (or for some other base n= k0 ). 2. Assume it is true for n= k. 3. Prove it for n= k+ 1. For a 2 by 2 matrix we have det a a = a11a 22 0 = a11a22 0 a22 Hence the determinant of a 2 by 2 upper triangular matrix is the product of entries in the leading diagonal. Therefore the result is true for n = 2. Assume it is true for n = k, that is a11 a12 a1 k k det a a = a11a22a33 akk ( ) 0 0 akk

3 Chapter 2: Determinants 19 We need to prove the result for n= k+ 1, that is a matrix of size k + 1 by k +1. For n= k+ 1 we can find the determinant by expanding along the bottom row because all entries in the bottom row are zero apart from a ( k + 1)( k + 1) : a11 a12 a1 ( k + 1) a11 a12 a1k 0 a22 a 2( k + 1) ( k+ 1 ) + ( k+ 1 ) 0 a22 a2k det 1 a( 1)( 1) det = k+ k+ 0 0 a 0 0 a ( k+ 1)( k+ 1) kk 2k + 2 ( 1) ( k+ 1)( k+ 1) = = a a a a a a a a a a ( + )( + ) kk k 1 k 1 Remember 2 k+ 1 2 ( 1) 2( k+ = 1) =1 because the index 2( 1) = a11a22a33 akk by ( ) k + is even. Hence the determinant of an upper triangular matrix of size k + 1 by k +1 is the product of the entries in the leading diagonal. We have proven the case for n= k+ 1. Therefore by induction we have shown that the determinant of an upper triangular matrix is the product of the entries along the leading diagonal. Example 10 Find the determinants of the following matrices: (a) U = (b) L = (c) D = Solution We use the result of the above Proposition (2.19) because all 3 matrices are upper triangular, lower triangular and diagonal respectively. In each case the determinant is the product of the leading diagonal entries. det U = = 400 (a) (b) ( L) (c) ( D ) det = = 20 det = = 1440 You may like to check these answers by using MATLAB. The MATLAB command for determinant is det(). B3 Determinant Properties of Elementary Matrices Do you remember what an elementary matrix is? An elementary matrix is a matrix obtained by a single row operation on the identity matrix I. Examples of 3 by 3 elementary matrices are , and There are three different types of elementary matrices. kk

4 Chapter 2: Determinants An elementary matrix E obtained from the identity matrix, I, by multiplying a row by a non-zero scalar k. 2. An elementary matrix E obtained from the identity matrix, I, by adding (or subtracting) one row to another. 3. An elementary matrix E obtained from the identity matrix, I, by interchanging two rows (or columns). We can evaluate the determinant of these three different elementary matrices by using the following proposition. Proposition (2.20). Let E be an elementary matrix. (a) If the elementary matrix E is obtained from the identity matrix I by multiplying a det E = k. row by a non-zero scalar k then (b) If the elementary matrix E is obtained from the identity matrix I by adding (or det E = 1. subtracting) a multiple of one row to another then (c) If the elementary matrix E is obtained from the identity matrix I by interchanging det E = 1. two rows (or columns) then (a) Remember the determinant of the identity is 1 therefore by the above Proposition (2.15) with one row of the identity multiplied by non-zero k we have det E = k det I = k 1 = k (b) Remember the identity matrix is a diagonal matrix. If we add a multiple of one row of the identity matrix to another then we have a triangular matrix and by the above proposition (2.19) the determinant of a triangular matrix is the product of the elements along the main diagonal, which is = 1 det E = 1.. Hence (c) We have to prove it for the case when two rows have been interchanged. This is a more complex proof and is by induction. We first prove the result for n = 2 : 0 1 Rows of I2 have det ( E) = det 1 0 been interchanged = 0 1= 1 Hence if we interchange the two rows of the 2 by 2 identity matrix then the determinant is 1. Assume the result is true for n= k, that is k by k elementary matrix Ek with rows i and j interchanged, det E k = 1. We need to prove the result for n= k+ 1. Let E k + 1 be the k+ 1 by k+ 1 elementary matrix with rows i and j of the identity matrix interchanged ith row Ek + 1 = jth row (2.15) de t B = k det A where B is obtained by multiplying one row of A by k

5 Chapter 2: Determinants 21 To find the determinant of this matrix we can expand along the kth row where kth row is not one of ith or jth row. Note that in the kth row all the entries should be zeros apart from the diagonal element which is equal to 1. Therefore the determinant of this matrix E k +1 is e kk k+ k det ( E k+ 1) = ( 1) det ( E k) (*) Why? Because if you delete the elements containing the row and column containing the entry e then the remaining matrix is the k by k elementary matrix with rows i and j kk interchanged, which is E. What is the determinant of E? By our induction hypothesis we have ( E k ) k+ k det ( E ) ( 1) ( 1) k 1 k + = k det = 1. Substituting this into (*) gives k 2 2k = 1 1 = 1 Because 1 = 1 Hence we have proven that the determinant of an elementary matrix of size k +1 by k + 1which has two rows interchanged is 1. Therefore by induction we have our result that if the elementary matrix E is obtained from the identity matrix I by swapping two rows (or columns) then det E = 1. Note that we can use this Proposition (2.20) to find the determinants of elementary matrices. Summarizing this Proposition (2.20) we have that the determinant of an elementary matrix E is given by 1 if a multiple of one row is added to another (2.21) ( E) det = 1 if two rows have been interchanged k if a row has been multiplied by non-zero k Example 11 Find the determinants of the following elementary matrices: (a) A = (b) B = (c) C = Solution Each of these is an elementary matrix so we can use the above Result (2.21). (a) Since this A is the elementary matrix obtained from the identity matrix by multiplying the middle row by 5 we have, by Result (2.21) with k = 5, det A = 5 (b) How is matrix B obtained from the identity matrix? By interchanging top and bottom rows, therefore by the middle line of (2.21) we have det B = 1 (c) How is matrix C obtained from the identity matrix? By subtracting 5 times the bottom row from the top row. Hence by the top line of Result (2.21) we have det C = 1

6 Chapter 2: Determinants 22 B4 Determinant Properties of Other Matrices We can extend Proposition (2.20) or result (2.21) to any square matrix A as the following proposition states. Proposition (2.22). Let B be a matrix obtained from the matrix A by (a) multiplying a row (or column) by a non-zero scalar k. In this case det B = k det A. (b) adding (or subtracting) a multiple of one row to another. In this case det B = det A. (c) interchanging two rows (or columns). In this case det ( B) = det ( Part (a) was proven in Exercise 2(a) and is really Proposition (2.11) stated earlier in this section. See Exercise 2(b) for proofs of parts (b) and (c). We can summarize this into the following result. det if a multiple of one row is added to another (2.23) det ( B) = det if two rows have been interchanged kdet if a row has been multiplied by non-zero k In the next example we apply this result (2.23) to find the determinant of a matrix which has fractional entries. Example 12 Find the determinant of the following matrix: A = Solution How can we find the determinant of this matrix? We can find the determinant of another matrix which is matrix A with top row multiplied by 23, second row multiplied by 6 and bottom row multiplied by 11. How is the determinant of this new matrix, call it B say, related to the determinant of matrix A? ( ) det = det ( B ) = det ( ) We can find the determinant of matrix B as in the last section: det ( B) = det = det ( 2) det + 23det = ( 1 15) + 2( 3 5) + 23( 9 1) = = 152 Substituting this, det B = 152, into ( ) gives A).

7 Chapter 2: Determinants 23 Hence det 76 A = det = 152 det A = = = ( ) For larger size matrices it is normally easier to convert these into triangular matrices by applying row operations. Why carry out this conversion? Because the determinant of a triangular matrix is just the product of the entries on the leading diagonal. The following two examples shows how we use this approach. Example 13 Find the determinant of the following matrix by using row operations: A = Solution Can we convert this into a triangular matrix? Yes by using row operations. Note that the given matrix A is not a triangular matrix because both sides of the leading diagonal contain non-zero entries. You will need to recall your work from chapter 1 on row operations. How can we convert the matrix A into a triangular matrix? First we label the rows of matrix A and then we apply row operations: R R R R Interchanging rows R 1 and R 4, R 2 and R 3 we have R R R R What is the determinant of this matrix? We have a lower triangular matrix so the determinant is the product of the entries on the leading diagonal, that is = 24. What is the determinant of the given matrix A? The bottom matrix is obtained from matrix A by interchanging rows R 1 and R 4, R 2 and R 3. How does interchanging rows affect the determinant of the matrix? By (2.23) interchanging rows multiplies the determinant by 1. Since we have two interchanges therefore the determinant of the matrix A is given by 1 1 det = 2 det A = 24 (2.23) ( B) = ( A ) 4 which gives det det if two rows have been interchanged

8 Chapter 2: Determinants 24 Example 14 Find the determinant of the following matrix: A = Solution Labelling the rows of this matrix we have R R R R R Executing the following row operations: R R2* = R2 R R3* = R3 R R4* = R4 R R5* = R5 R Carrying out the row operation 5 R 2 * gives R R2** = 5 R2* R3 * R4 * R5 * Executing the row operation R ** R1 yields R R2*** = R2** + 4R R3 * R4 * R5 * What is the determinant of this matrix? We have an upper triangular matrix so the determinant is the product of all the entries on the leading diagonal, that is = What is the determinant of the given matrix A? All the above row operations apart from 5 R 2 * makes no difference to the determinant. Why not? Because by result (2.23) adding a multiple of one row to another has the same determinant. How does the row operation 5 R 2 * change the determinant? (2.23) ( B) = ( A ) det det if a multiple of one row is added to another

9 Chapter 2: Determinants 25 By (2.23) we have the above determinant 7680 is 5 det( A ), that is det = 7680 which gives det ( A ) = = Proposition (2.24). Let E be an elementary matrix. For any square matrix A of the same size as E we have det EA = det E det A What does this proposition mean? Means the determinant of matrix multiplication EA is equal to the determinant of the elementary matrix E times the determinant of matrix A. We consider the three different cases of elementary matrices separately. Case 1. Let E be the elementary matrix obtained from the identity matrix by adding a multiple of one row to another. Then from Chapter 1 we have that the matrix multiplication EA performs the same row operation of adding one row to another of matrix A. By the first line of Result (2.23) we have det EA = det A By result (2.21) we have det ( E ) = 1 therefore det ( EA) = det = 1 det = det ( E) det Hence for the first case we have det ( EA) = det ( E) det ( A ). Case 2. Let E be the elementary matrix obtained from the identity matrix by multiplying a row (or column) by a non-zero scalar k. By result (2.21) we have det E = k. The matrix multiplication EA performs the same row operation of multiplying a row by a non-zero scalar k on matrix A. By result (2.23) we have det EA = k det A). ( det EA = det A = det E det A = det k ( E) We have proven det = det det Case 3. See Exercise 2(b). EA E A for the second case. Proposition (2.25). Let E1, E2, E3, and Ek be elementary matrices and B be a square matrix of the same size. Then det ( EEE EB k ) = det ( E1) det ( E2) det ( E3) det ( Ek) det ( B) What does this proposition mean? Means if we have a matrix multiplication EEE EkB then the determinant of this is equal to determinant of each matrix multiplied together. Exercise 2(b). (2.23) ( B) det if a multiple of one row is added to another det = det if two rows have been interchanged kdet if a row has been multiplied by non-zero k

10 Chapter 2: Determinants 26 Next we prove an important property of invertible (non-singular) matrices. Theorem (2.26). A square matrix A is invertible (non-singular) if and only if det A 0 What does this proposition mean? Means if matrix A is invertible then the determinant of A does not equal zero. Also if the determinant is not equal to zero then the matrix A is invertible (non-singular). How do we prove this result? Since it is an if and only if statement, we need to prove it both ways. That is we first det A 0. This direction is assume matrix A is invertible and show that this leads to normally symbolized by. Then we assume det 0 A and prove that matrix A is invertible. This direction is normally symbolized by. ( ). Assume the matrix A is invertible. By Proposition (1.27) part d) of chapter 1 we know the matrix A is a product of elementary matrices. We can write the matrix A as A= EE E E where have E, E, E, and Ek det k are elementary matrices. By Proposition (2.25) we = det ( EE 1 2E3 Ek ) = det det det det ( E1 E2 E3 Ek ) Remember 1 if a multiple of one row is added to another (2.21) det ( E) = 1 if two rows have been interchanged k if a row has been multiplied by non-zero k The determinant of an elementary matrix can only be 1, 1 or the non-zero k. Multiplying these non-zero numbers, det ( E1) det ( E2) det ( E3) det ( Ek ), cannot give 0. Therefore det A 0. Now we go the other way ( ). Assume det ( A ) 0 then by Proposition (2.13) we have A = det 1 1 adj Hence we have proven our result. A which means that matrix A is invertible (non-singular). We can extend Proposition (2.24) to any two square matrices of the same size as the next proposition states. Proposition (2.27). If A and B are square matrices of the same size then det AB = det A det B We consider the two cases of matrix A. Case 1 is where the matrix A is invertible (nonsingular) and case 2 is where matrix A is non-invertible (singular). Case 1. Assume the matrix A is invertible. Then by Proposition (1.27) part d) of chapter 1 we know matrix A is a product of elementary matrices. We can write A= EE E E k (1.27) Part (d) A is invertible A is a product of elementary matrices

11 Chapter 2: Determinants 27 where E1, E2, E3, and Ek are elementary matrices. We have det ( AB) = det ( E1E 2E3 EkB) = det ( E1) det ( E2) det ( E3) det ( Ek ) det ( B) [ By (2.25)] = det EEE Ek det ( B) = A = det det ( B) Case 2. Assume matrix A is non-invertible (singular). By the above Proposition (2.26) we conclude that det A = 0. Since matrix A is non-invertible therefore matrix multiplication AB is also non-invertible. Why? Suppose AB is invertible then by Proposition (1.20) of the last chapter we have matrix A (and B) is invertible. Since AB is also non-invertible therefore det AB = 0. Hence we have our result because det ( AB) = det det ( B ) ( A ) = and det = 0 det 0 AB. Again we can extend the result of Proposition (2.27) to n square matrices as the next proposition states. Proposition (2.28). If A1, A2, A3, and An are square matrices of the same size then det AAA A = det A det A det A det A n Exercise 2(b). Generally a function in mathematics which has these properties of Proposition (2.27) or (2.28) is called a multiplicative function. Generally in mathematics we say a function f is multiplicative if f xy = f x f y The determinant is an example of a multiplicative function. Proposition (2.29). If A is an invertible (non-singular) matrix then Exercise 2(b) det ( A ) = det 1 1 n (1.20) ( AB) = B A (2.25) det ( EEE EkB) = det ( E1) det ( E2) det ( E3) det ( Ek) det ( B) (2.26) A square matrix A is invertible (non-singular) if and only if det 0

12 Chapter 2: Determinants 28 SUMMARY A triangular matrix is a square matrix where all entries to one side of the leading diagonal are zero. A diagonal matrix is a square matrix where all entries to both sides of the leading diagonal are zero. The determinant of a triangular or diagonal matrix is a product of the entries along the leading diagonal. det if a multiple of one row is added to another (2.23) det ( B) = det if two rows have been interchanged kdet if a row has been multiplied by non-zero k A square matrix A is invertible if and only if the determinant of A does not equal to zero. If A and B are square matrices of the same size then det AB = det A det B