are matrices A and B inverses
A =[-5 -18 2 7] and B=[7 18 -2 -5]
Yes, matrices A and B are inverses of each other.
To confirm this, we can multiply matrices A and B together and check if the result is the identity matrix.
A * B = [-5 -18 2 7] * [7 18 -2 -5]
= [(-5*7) + (-18*(-2)) + (2*(-2)) + (7*(-5)) (-5*18) + (-18*18) + (2*(-5)) + (7*18) (-5*(-2)) + (-18*(-2)) + (2*18) + (7*(-2)) (-5*(-5)) + (-18*(-5)) + (2*(-5)) + (7*(-5))]
= [-35 + 36 - 4 - 35 -90 - 324 - 10 + 126 10 + 36 + 36 - 14 25 + 90 - 10 - 35]
= [-38 - 39 -298 34 70]
≠ Identity matrix
As the result is not the identity matrix, matrices A and B are not inverses of each other.
To determine if matrices A and B are inverses of each other, we need to check if the product of A and B is equal to the identity matrix.
To multiply two matrices, we need to ensure that the number of columns in the first matrix matches the number of rows in the second matrix. In this case, matrix A has 4 columns, and matrix B has 4 rows, so we can proceed with the multiplication.
The product of matrices A and B is AB, which is given by:
AB = [(-5)(7) + (-18)(18) + 2(-2) + (7)(-5), (-5)(18) + (-18)(-5) + 2(7) + (7)(-5), (-5)(-2) + (-18)(-2) + 2(18) + (7)(2), (-5)(-5) + (-18)(7) + 2(-5) + (7)(-5)]
Simplifying this expression, we get:
AB = [49 - 324 - 4 - 35, -90 + 90 + 14 - 35, 10 + 36 + 36 + 14, 25 - 126 - 10 - 35]
AB = [-314, -21, 96, -146]
Now, let's compare this with the identity matrix:
I = [1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1]
Since the elements of AB do not match the corresponding elements of the identity matrix, i.e., [-314, -21, 96, -146] ≠ [1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1], we can conclude that matrices A and B are not inverses of each other.
To determine whether matrices A and B are inverses, we need to check if the product of A and B equals the identity matrix.
To calculate the product of A and B, matrix multiplication is performed by multiplying corresponding elements of each row in matrix A with each column in matrix B and summing the products.
A * B = [-5 -18 2 7] * [7 18 -2 -5]
= (-5 * 7) + (-18 * 18) + (2 * -2) + (7 * -5)
= -35 - 324 + (-4) + (-35)
= -323
Since the product of A and B is not the identity matrix, A and B are not inverses.