Find the inverse of matrix A.
.6 -2
A= 6 -18
To find the inverse of a matrix, you can follow these steps:
Step 1: Write down the matrix A
A = [[0.6, -2], [6, -18]]
Step 2: Calculate the determinant of A, denoted as |A|
|A| = (0.6 * -18) - (6 * -2)
Step 3: Check if the determinant |A| is nonzero. If |A| = 0, then the matrix A does not have an inverse. If |A| is not zero, it means that A has an inverse.
In this case, calculating |A|, we have:
|A| = (0.6 * -18) - (6 * -2)
= -10.8 + 12
= 1.2
Since |A| is not equal to zero, matrix A has an inverse.
Step 4: Calculate the adjugate of A, denoted as adj(A)
The adjugate of matrix A is obtained by transposing the matrix of cofactors of A. To find the adjugate:
adj(A) = [[cofactor(A[0][0]), cofactor(A[0][1])], [cofactor(A[1][0]), cofactor(A[1][1])]]
To find the cofactor of an element, you need to calculate the determinant of the submatrix obtained by deleting the row and column containing that element, and then multiply it by (-1)^(row index + column index)
For A[0][0] = 0.6:
cofactor(A[0][0]) = det(A[1][1])
= -18
For A[0][1] = -2:
cofactor(A[0][1]) = det(A[1][0])
= 6
For A[1][0] = 6:
cofactor(A[1][0]) = det(A[0][1])
= -2
For A[1][1] = -18:
cofactor(A[1][1]) = det(A[0][0])
= 0.6
Therefore, the adjugate of matrix A is:
adj(A) = [[-18, 6], [-2, 0.6]]
Step 5: Calculate the inverse of A, denoted as A^(-1)
To calculate the inverse of A, you need to divide the adjugate of A by the determinant of A:
A^(-1) = (1/|A|) * adj(A)
Since |A| = 1.2, the inverse of matrix A is:
A^(-1) = [[-18/1.2, 6/1.2], [-2/1.2, 0.6/1.2]]
= [[-15, 5], [-1.67, 0.5]]
So, the inverse of matrix A is:
A^(-1) = [[-15, 5], [-1.67, 0.5]]