Finding the Multiplicative Inverse of a 3 × 3 Matrix Find the multiplicative inverse of A = [1 -1 1 0 -2 1 -2 -3 0].

Question

Finding the Multiplicative Inverse of a 3 × 3 Matrix

Find the multiplicative inverse of

[latex]A=\left[\begin{array}{rrr}1 & -1 & 1 \0 & -2 & 1 \-2 & -3 & 0\end{array} ight][/latex].

Accepted Answer

Step 1 Form the augmented matrix [latex]\left[A \mid I_3 ight][/latex].

Step 2 Perform row operations on [latex]\left[A \mid I_3 ight][/latex] to obtain a matrix of the form [latex]\left[I_3 \mid B ight][/latex]. To the left of the vertical dividing line, we want 1 s down the diagonal from upper left to lower right and 0s elsewhere.

[latex]\left[\begin{array}{rrr|rrr}1 & -1 & 1 & 1 & 0 & 0 \0 & -2 & 1 & 0 & 1 & 0 \-2 & -3 & 0 & 0 & 0 & 1\end{array} ight]\overset{ ext { Replace row } 3 ext { by } 2 R_1+R_3 ext {. }}{\longrightarrow }[/latex]

[latex]\left[\begin{array}{lll|lll}1 & -1 & 1 & 1 & 0 & 0 \0 & -2 & 1 & 0 & 1 & 0 \0 & -5 & 2 & 2 & 0 & 1\end{array} ight]\overset{-\frac{1}{2} R_2}{\longrightarrow } [/latex]

[latex]\left[\begin{array}{rrr|rrr}1 & -1 & 1 & 1 & 0 & 0 \0 & 1 & -\frac{1}{2} & 0 & -\frac{1}{2} & 0 \0 & -5 & 2 & 2 & 0 & 1\end{array} ight]\overset{ ext{Replace row 1 by 1} R_2+R_1.\ ext{Replace row 3 by 5} R_2+R_3.}{\longrightarrow }[/latex]

[latex]\left[\begin{array}{rrr|rrr}1 & 0 & \frac{1}{2} & 1 & -\frac{1}{2} & 0 \0 & 1 & -\frac{1}{2} & 0 & -\frac{1}{2} & 0 \0 & 0 & -\frac{1}{2} & 2 & -\frac{5}{2} & 1\end{array} ight]\overset{-2 R_3}{\longrightarrow } [/latex]

Step 3 Matrix B is [latex]A^{-1}[/latex]. The last matrix shown at the bottom of the previous page is in the form [latex]\left[I_3 \mid B ight][/latex]. The multiplicative identity matrix is on the left of the vertical bar. Matrix B, the multiplicative inverse of A, is on the right. Thus, the multiplicative inverse of A is

[latex]A^{-1}=\begin{bmatrix} 3 & -3&1 \ -2 & 2&-1 \-4&5&-2\end{bmatrix}[/latex]

Step 4 Verify the result by showing that [latex]AA^{-1} = I_3[/latex] and [latex]A^{-1}A = I_3[/latex]. Try confirming the result by multiplying A and A[latex]^{-1}[/latex] to obtain [latex]I_3[/latex]. Do you obtain [latex]I_3[/latex] if you reverse the order of the multiplication?

Question 8.4.4: Finding the Multiplicative Inverse of a 3 × 3 Matrix Find th...

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