# Find the inverse of each of the following matrices. begin{bmatrix}1 & 0&1-1&1&1 -1&-2&-3 end{bmatrix}

Find the inverse of each of the following matrices.
$\left[\begin{array}{ccc}1& 0& 1\\ -1& 1& 1\\ -1& -2& -3\end{array}\right]$
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Nathalie Redfern
Step 1
Given matrix is
$A=\left[\begin{array}{ccc}1& 0& 1\\ -1& 1& 1\\ -1& -2& -3\end{array}\right]$
firstly, we will find all cofactors by using the formula
${A}_{ij}=\left(-1{\right)}^{i+j}{M}_{ij}$
where ${M}_{ij}$ are minors.
${A}_{11}=\left(-1{\right)}^{1+1}\left(-3+2\right)=-1$
${A}_{12}=\left(-1{\right)}^{1+2}\left(3+1\right)=-4$
${A}_{13}=\left(-1{\right)}^{1+3}\left(2+1\right)=3$
${A}_{21}=\left(-1{\right)}^{2+1}\left(0+2\right)=-2$
${A}_{22}=\left(-1{\right)}^{2+2}\left(-3+1\right)=-2$
${A}_{23}=\left(-1{\right)}^{2+3}\left(-2+0\right)=2$
${A}_{31}=\left(-1{\right)}^{3+1}\left(0-1\right)=-1$
${A}_{32}=\left(-1{\right)}^{3+2}\left(1+1\right)=-2$
${A}_{33}=\left(-1{\right)}^{3+3}\left(1+0\right)=1$
Now, Adjoint A matrix is transpose of cofactors matrix.
$AdjA={\left[\begin{array}{ccc}-1& -4& 3\\ -2& -2& 2\\ -1& -2& 1\end{array}\right]}^{T}=\left[\begin{array}{ccc}-1& -2& -1\\ -4& -2& -2\\ 3& 2& 1\end{array}\right]$
Step 2
Now, we ill evaluate the determinant of matrix A
$|A|={a}_{11}{A}_{11}+{a}_{12}{A}_{12}+{a}_{13}{A}_{13}$
$=1\left(-1\right)+0\left(-4\right)+1\left(3\right)=2$
Since, determinant of A is non-zero, therefore, inverse matrix of A exists and is obtained as
${A}^{-1}=\frac{1}{|A|}AdjA$
${A}^{-1}=\frac{1}{2}\left[\begin{array}{ccc}-1& -2& -1\\ -4& -2& -2\\ 3& 2& 1\end{array}\right]$
Step 3
Ans:
${A}^{-1}=\frac{1}{2}\left[\begin{array}{ccc}-1& -2& -1\\ -4& -2& -2\\ 3& 2& 1\end{array}\right]$
Jeffrey Jordon