Consider the following Matrix:
A= [-2 0 1
-1 -2 1
1 -1 0 ]
Choose the correct description of
A.
Find A−1 if it exists.
Multiple Choice homework question:
1) A is Nonsingular; that is, it has an inverse.
A^-1=
2) A is singular; that is, its inverse doesn't exist.
Thanks!
To determine whether matrix A has an inverse (A^-1) or not, we need to calculate the determinant of matrix A. If the determinant is nonzero, then A has an inverse; otherwise, it does not.
To calculate the determinant of matrix A, you can use the formula for a 3x3 matrix:
det(A) = (-2 * (-2*0 - 1*(-1))) - (0 * (-1*0 - 1*1)) + (1 * (-1*(-1) - (-2)*1))
= (-2 * (-2)) - (0 * (-1)) + (1 * (1 - (-2)))
= 4 + 0 + 3
= 7
Since the determinant of matrix A is nonzero (7 is not equal to zero), matrix A is nonsingular. This means that A has an inverse (A^-1).
Therefore, the correct description is 1) A is Nonsingular; that is, it has an inverse.
To find the inverse of matrix A, you can use the formula:
A^-1 = (1/det(A)) * adj(A)
Where adj(A) represents the adjugate of matrix A.
The adjugate of matrix A is obtained by taking the transpose of the cofactor matrix of A. To find the cofactor matrix of A, follow these steps:
Step 1: Calculate the cofactor of each element in matrix A. The cofactor of an element a_ij is given by (-1)^(i+j) * det(A_ij), where A_ij represents the submatrix obtained by removing the i-th row and j-th column from matrix A.
The cofactor matrix of A is given by:
C = [C_11 C_12 C_13
C_21 C_22 C_23
C_31 C_32 C_33]
Where C_ij represents the cofactor of element a_ij.
Step 2: Take the transpose of matrix C to obtain the adjugate matrix of A.
Once you have the adjugate matrix, multiply it by (1/det(A)) to find the inverse matrix A^-1.
By following these steps, you should be able to find A^-1 for the given matrix A = [-2 0 1; -1 -2 1; 1 -1 0].