In Exercises 13 - 18, use the fact that if a b d - b A = then A^(-1) = 1/(ad-bc) to find the inverse of c d - c a each matrix, if possible. Check that AA^(-1) = I_2 and A^(-1)A = I_2. 10 - 2 A = - 5 1

Verified step by step guidance

1. Identify the elements of the matrix A: a = 10, b = -2, c = -5, and d = 1.
2. Calculate the determinant of A using the formula $\mathrm{ad}-\mathrm{bc}$. Substitute the values: $(10)(1)\; -\; (-2)(-5)$.
3. Check if the determinant is not zero. If it is zero, the inverse does not exist. If it is not zero, proceed to find the inverse.
4. Use the formula for the inverse matrix: . Substitute the values of a, b, c, and d.
5. Multiply the scalar $\backslash frac\{1\}\{ad\; -\; bc\}$ by each element of the matrix to get the inverse matrix.

Key Concepts

Matrix Inverse

The inverse of a matrix A, denoted A⁻¹, is a matrix that when multiplied by A results in the identity matrix I. For a 2x2 matrix, the inverse exists only if the determinant (ad - bc) is nonzero. The formula for the inverse involves swapping and negating elements and dividing by the determinant.

Determinant of a 2x2 Matrix

The determinant of a 2x2 matrix A = [[a, b], [c, d]] is calculated as ad - bc. It is a scalar value that indicates whether the matrix is invertible. If the determinant is zero, the matrix does not have an inverse, meaning it is singular.

Verification of Matrix Inverse

To confirm that a matrix B is the inverse of A, multiply A by B and B by A. Both products should equal the 2x2 identity matrix I₂. This verification ensures the correctness of the inverse calculation and the invertibility of the matrix.