Q2. - Algebra 2

Question

Explain how to find the inverse of a$\mathbf{2}\mathbf{\times }\mathbf{2}$ matrix.

Step-by-Step Solution

Answer

The inverse of$\mathbf{2}\mathbf{\times }\mathbf{2}$ matrix $\mathbf{A}\mathbf{=}\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$is given by${\mathbf{A}}^{-1}\mathbf{=}\frac{1}{\mathrm{ad}-\mathrm{bc}}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$ and $\mathbf{ad}\mathbf{-}\mathbf{bc}\ne \mathbf{0}$.

Step 1 ­- Description of step.

A$2\times 2$ matrix means the matrix is having two rows and two columns.

Step 2 ­- Definition of inverse of matrix.

A square matrixB is said to be an inverse of the square matrixA if$\mathbf{AB}\mathbf{=}\mathbf{BA}\mathbf{=}\mathbf{I}$ where I is an identity matrix of the same order as that of matrix Aor B.

Step 3 ­- Description of step.

The inverse of$2\times 2$ matrix $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$is given by $A^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$and $ad-bc\ne 0$.