# Determine whether the given matrices are inverses of each other. A=begin{bmatrix} 8 & 3 &-4 -6 & -2 &3-3&1&1 end{bmatrix} text{ and } B=begin{bmatrix} -1 & -1 &-1 3 & 4 &00&1&-2 end{bmatrix}

Determine whether the given matrices are inverses of each other.
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Derrick
Step 1
Here we are given two matrices:
$AB=\left[\begin{array}{ccc}8& 3& -4\\ -6& -2& 3\\ -3& 1& 1\end{array}\right]\left[\begin{array}{ccc}-1& -1& -1\\ 3& 4& 0\\ 0& 1& -2\end{array}\right]$
To show that the given matrices are multiplicative inverses of each other.
Step 2
Multiply AB and BA and if both products equal the identity, then the two matrices are inverses of each other:
Find AB and BA:
$AB=\left[\begin{array}{ccc}8& 3& -4\\ -6& -2& 3\\ -3& 1& 1\end{array}\right]\left[\begin{array}{ccc}-1& -1& -1\\ 3& 4& 0\\ 0& 1& -2\end{array}\right]$
$=\left(\begin{array}{ccc}8\left(-1\right)+3\cdot 3+\left(-4\right)\cdot 0& 8\left(-1\right)+3\cdot 4+\left(-4\right)\cdot 1& 8\left(-1\right)+3\cdot 0+\left(-4\right)\left(-2\right)\\ \left(-6\right)\left(-1\right)+\left(-2\right)\cdot 3+3\cdot 0& \left(-6\right)\left(-1\right)+\left(-2\right)\cdot 4+3\cdot 1& \left(-6\right)\left(-1\right)+\left(-2\right)\cdot 0+3\cdot \left(-2\right)\\ \left(-3\right)\left(-1\right)+1\cdot 3+1\cdot 0& \left(-3\right)\left(-1\right)+1\cdot 4+1\cdot 1& \left(-3\right)\left(-1\right)+1\cdot 0+1\cdot \left(-2\right)\end{array}\right)$
$=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 6& 9& 1\end{array}\right)$
Step 3
Find the product BA:
$BA=\left[\begin{array}{ccc}-1& -1& -1\\ 3& 4& 0\\ 0& 1& -2\end{array}\right]\left[\begin{array}{ccc}8& 3& -4\\ -6& -2& 3\\ -3& 1& 1\end{array}\right]$
$=\left[\begin{array}{ccc}\left(-1\right)\cdot 8+\left(-1\right)\left(-6\right)+\left(-1\right)\left(-3\right)& \left(-1\right)\cdot 3+\left(-1\right)\left(-2\right)+\left(-1\right)\cdot 1& \left(-1\right)\left(-4\right)+\left(-1\right)\cdot 3+\left(-1\right)1\\ 3\cdot 8+4\left(-6\right)+0\left(-3\right)& 3\cdot 3+4\left(-2\right)+0\cdot 1& 3\left(-4\right)+4\cdot 3+0\cdot 1\\ 0\cdot 8+1\left(-6\right)+\left(-2\right)\left(-3\right)& 0\cdot 3+1\left(-2\right)+\left(-2\right)\cdot 1& 0\left(-4\right)+1\cdot 3+\left(-2\right)1\end{array}\right]$
$=\left[\begin{array}{ccc}1& -2& 0\\ 0& 1& 0\\ 0& -4& 1\end{array}\right]$
Step 4
So, the product of A and B matrices are not identity matrix. so, that the matrices are not inverse of each other.
Jeffrey Jordon