Use Elementary Operations (a) Compute the determinant of the matrix A, reducing
ID: 3039553 • Letter: U
Question
Use Elementary Operations (a) Compute the determinant of the matrix A, reducing the matrix to a simpler matrix (usually triangular), by elementary operations: A = (1 1 1 1 -3 -2 -2 -2 2 2 4 5 2 2 2 3) (b) Compute the de of the matrix A, reducing the matrix to a simpler matrix (usually triangular), by elementary operations: A = (0 0 0 1 0 1 1 1 0 0 1 1 1 1 1 1) (c) Compute the determinant of the matrix A, reducing the matrix to a simpler matrix (usually triangular), by elementary operations: A = (0 0 0 1 0 1 1 1 0 0 1 1 squareroot 2 squareroot 2 squareroot 2 squareroot 2) (d) Compute the determinant of the matrix A, reducing the matrix to a simpler matrix (usually triangular), by elementary operations: A = (squareroot 2 squareroot 2 squareroot 2 1 + squareroot 2 0 1 1 1 0 0 1 1 squareroot 2 squareroot 2 squareroot 2 squareroot 2) (e) Compute the determinant of the matrix A, reducing the matrix to a simpler matrix (usually triangular), by elementary operations: A = (0 0 0 1 0 pi pi pi 0 0 1 1 squareroot 2 squareroot 2 squareroot 2 squareroot 2)Explanation / Answer
2. We know that
(a). To compute the determinant of the given matrix, A (say), we will reduce it to its RREF as under:
Add 3 times the 1st row to the 2nd row; Add -2 times the 1st row to the 3rd row
Add -2 times the 1st row to the 4th row; Multiply the 3rd row by ½
Add -3/2 times the 4th row to the 3rd row; Add -1 times the 4th row to the 2nd row
Add -1 times the 4th row to the 1st row; Add -1 times the 3rd row to the 2nd row
Add -1 times the 3rd row to the 1st row; Add -1 times the 2nd row to the 1st row
Then the RREF of A is
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
It may be observed that only the 4th row operation affects the value of det(A), by scaling its value by ½. Further, the determinant of an upper triangular matrix is the product of the diagonal entries so that the value of the determinant of the RREF of A is 1 and hence the value of det(A) is 2.
(b). To compute the determinant of the given matrix, A (say), we will reduce it to its RREF as under:
Interchange the 1st row and the 4th row; Add -1 times the 4th row to the 3rd row
Add -1 times the 4th row to the 2nd row; Add -1 times the 4th row to the 1st row
Add -1 times the 3rd row to the 2nd row; Add -1 times the 3rd row to the 1st row
Add -1 times the 2nd row to the 1st row
Then the RREF of A is
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
It may be observed that the 1st row operation changes the sign of det(A). The sum total of these row operations is a change in the value of det(A) to –det(A). Further, the determinant of an upper triangular matrix is the product of the diagonal entries so that the value of the determinant of the RREF of A is 1 and hence the value of det(A) is -1.
(c ). Here, the matrix A may be reduced to its RREF by the same row operations as in (b) above except that the 1st row operation is “multiply the 4th row by 1/2. Further, the RREF of A is again I4 as in (b) above. It may be observed that the 1st row operation scales det(A) by 1/2. The 2nd row operation changes the sign of det(A). The sum total of these row operations is a change in the value of det(A) to –(1/2)det(A). Further, the determinant of an upper triangular matrix is the product of the diagonal entries so that the value of the determinant of the RREF of A is 1 and hence the value of det(A) is -2.
(d). Here, the matrix A may be reduced to its RREF by the same row operations as in (c) above except that the 1st row operation is “add -1 time the 4th row to the 1st row. Further, the RREF of A is again I4 as in (c) above. It may be observed that the 2nd row operation scales det(A) by 1/2. The 3rd row operation changes the sign of det(A). The sum total of these row operations is a change in the value of det(A) to –(1/2)det(A). Further, the determinant of an upper triangular matrix is the product of the diagonal entries so that the value of the determinant of the RREF of A is 1 and hence the value of det(A) is -2.
(e ). Here, the matrix A may be reduced to its RREF by the same row operations as in (c) above except that the 1st row operation is “multiply 2nd row by 1/. Further, the RREF of A is again I4 as in (c) above. It may be observed that the 2nd row operation scales det(A) by 1/2. The 3rd row operation changes the sign of det(A). The sum total of these row operations is a change in the value of det(A) to-(1/2)det(A). Further, the determinant of an upper triangular matrix is the product of the diagonal entries so that the value of the determinant of the RREF of A is 1 and hence the value of det(A) is -2.
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
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