Find the inverse, if it exists, for each matrix. [2x2 matrix]

Verified step by step guidance

1. Step 1: Understand the concept of an inverse matrix. For a 2x2 matrix $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$, the inverse $A^{-1}$ exists if and only if the determinant of $A$ is non-zero.
2. Step 2: Calculate the determinant of the matrix $A$. The determinant is given by $\text{det}(A)=ad-bc$.
3. Step 3: Check if the determinant is non-zero. If $\text{det}(A)\ne 0$, the inverse exists. If $\text{det}(A)=0$, the matrix does not have an inverse.
4. Step 4: If the inverse exists, use the formula for the inverse of a 2x2 matrix: $A^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$.
5. Step 5: Substitute the values of $a,b,c,$ and $d$ from the given matrix into the formula to find the inverse matrix.

Key Concepts

Matrix Inversion

Matrix inversion is the process of finding a matrix that, when multiplied with the original matrix, yields the identity matrix. For a 2x2 matrix, the inverse can be calculated using the formula A^(-1) = (1/det(A)) * adj(A), where det(A) is the determinant and adj(A) is the adjugate of the matrix. An inverse exists only if the determinant is non-zero.

Determinant

The determinant is a scalar value that provides important information about a matrix, including whether it is invertible. For a 2x2 matrix represented as [[a, b], [c, d]], the determinant is calculated as det(A) = ad - bc. If the determinant equals zero, the matrix does not have an inverse, indicating that the rows (or columns) are linearly dependent.

Adjugate of a Matrix

The adjugate (or adjoint) of a matrix is a matrix whose elements are the cofactors of the original matrix, transposed. For a 2x2 matrix, the adjugate can be found by swapping the elements on the main diagonal and changing the signs of the off-diagonal elements. The adjugate is used in the formula for finding the inverse, highlighting its role in matrix operations.