Find (a) det(A), (b) the matrix of cofactors MC,(c) adj (A), and, if possible, (d) A^-1. A=[

step 1 this is our question number 40 first of all find the determinant of the matrix so here minus 1 m

Find (a) det(A), (b) the matrix of cofactors MC,(c) adj (A),...

Find (a) $\operatorname{det}(A),$ (b) the matrix of cofactors $M_{C},(\mathrm{c})$ adj $(A),$ and, if possible, $(\mathrm{d}) A^{-1}.$
$$A=\left[\begin{array}{rr}
-1 & -2 \\
4 & 1
\end{array}\right]$$

Key Concepts

Matrix Inverse

The inverse of a matrix, when it exists, is a matrix that, when multiplied by the original matrix, yields the identity matrix. The inverse can be computed using the formula A^(-1) = (1/det(A)) multiplied by the adjugate of A, provided that the determinant of A is nonzero, which guarantees the matrix is invertible.

Adjugate Matrix

The adjugate matrix, also known as the adjoint, is the transpose of the matrix of cofactors. It plays an important role in finding the inverse of a matrix. The property that A^(-1) = (1/det(A)) × adj(A) is fundamental for computing the inverse when the determinant is nonzero.

Determinant

The determinant is a scalar value that is computed from the elements of a square matrix and provides important information about the matrix properties, such as invertibility and volume scaling. In a 2×2 matrix, the determinant is calculated using the formula ad ? bc, and a nonzero determinant indicates that the matrix is invertible.

Cofactors

Cofactors are numbers associated with each element of a matrix that are used in the computation of the determinant, the adjugate matrix, and the inverse of a matrix. Each cofactor is obtained by multiplying the minor (the determinant of the submatrix formed by removing the row and column of the given element) by (-1)^(i+j), where i and j denote the row and column positions.

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