Matrix Inverses — Practice Problems and Worked Solutions

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18 worked problems

Every step is generated and checked by a computer algebra system.

1. 1.

   Find the matrix inverse.

   $$A=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right],A^{-1}=?$$

   Answer: $\left[\begin{array}{cc}1& -1\\ 0& 1\end{array}\right]$

   Step-by-step solution

   1. $$A^{-1}=\frac{1}{\det A}\mathrm{adj}(A)$$
   2. $$A^{-1}=\left[\begin{array}{cc}1& -1\\ 0& 1\end{array}\right]$$
   3. $$A{A}^{-1}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$$
2. 2.

   Find the matrix inverse.

   $$A=\left[\begin{array}{cc}2& 1\\ 1& 1\end{array}\right],A^{-1}=?$$

   Answer: $\left[\begin{array}{cc}1& -1\\ -1& 2\end{array}\right]$

   Step-by-step solution

   1. $$A^{-1}=\frac{1}{\det A}\mathrm{adj}(A)$$
   2. $$A^{-1}=\left[\begin{array}{cc}1& -1\\ -1& 2\end{array}\right]$$
   3. $$A{A}^{-1}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$$
3. 3.

   Find the matrix inverse.

   $$A=\left[\begin{array}{cc}3& -1\\ 2& 1\end{array}\right],A^{-1}=?$$

   Answer: $\left[\begin{array}{cc}\frac{1}{5}& \frac{1}{5}\\ -\frac{2}{5}& \frac{3}{5}\end{array}\right]$

   Step-by-step solution

   1. $$A^{-1}=\frac{1}{\det A}\mathrm{adj}(A)$$
   2. $$A^{-1}=\left[\begin{array}{cc}\frac{1}{5}& \frac{1}{5}\\ -\frac{2}{5}& \frac{3}{5}\end{array}\right]$$
   3. $$A{A}^{-1}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$$
4. 4.

   Find the matrix inverse.

   $$A=\left[\begin{array}{cc}1& 2\\ 3& 5\end{array}\right],A^{-1}=?$$

   Answer: $\left[\begin{array}{cc}-5& 2\\ 3& -1\end{array}\right]$

   Step-by-step solution

   1. $$A^{-1}=\frac{1}{\det A}\mathrm{adj}(A)$$
   2. $$A^{-1}=\left[\begin{array}{cc}-5& 2\\ 3& -1\end{array}\right]$$
   3. $$A{A}^{-1}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$$
5. 5.

   Find the matrix inverse.

   $$A=\left[\begin{array}{cc}4& 1\\ -2& 1\end{array}\right],A^{-1}=?$$

   Answer: $\left[\begin{array}{cc}\frac{1}{6}& -\frac{1}{6}\\ \frac{1}{3}& \frac{2}{3}\end{array}\right]$

   Step-by-step solution

   1. $$A^{-1}=\frac{1}{\det A}\mathrm{adj}(A)$$
   2. $$A^{-1}=\left[\begin{array}{cc}\frac{1}{6}& -\frac{1}{6}\\ \frac{1}{3}& \frac{2}{3}\end{array}\right]$$
   3. $$A{A}^{-1}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$$
6. 6.

   Find the matrix inverse.

   $$A=\left[\begin{array}{cc}2& -3\\ 1& -1\end{array}\right],A^{-1}=?$$

   Answer: $\left[\begin{array}{cc}-1& 3\\ -1& 2\end{array}\right]$

   Step-by-step solution

   1. $$A^{-1}=\frac{1}{\det A}\mathrm{adj}(A)$$
   2. $$A^{-1}=\left[\begin{array}{cc}-1& 3\\ -1& 2\end{array}\right]$$
   3. $$A{A}^{-1}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$$
7. 7.

   Find the matrix inverse.

   $$A=\left[\begin{array}{cc}1& 0\\ 2& 1\end{array}\right],A^{-1}=?$$

   Answer: $\left[\begin{array}{cc}1& 0\\ -2& 1\end{array}\right]$

   Step-by-step solution

   1. $$A^{-1}=\frac{1}{\det A}\mathrm{adj}(A)$$
   2. $$A^{-1}=\left[\begin{array}{cc}1& 0\\ -2& 1\end{array}\right]$$
   3. $$A{A}^{-1}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$$
8. 8.

   Find the matrix inverse.

   $$A=\left[\begin{array}{cc}3& 1\\ 1& 1\end{array}\right],A^{-1}=?$$

   Answer: $\left[\begin{array}{cc}\frac{1}{2}& -\frac{1}{2}\\ -\frac{1}{2}& \frac{3}{2}\end{array}\right]$

   Step-by-step solution

   1. $$A^{-1}=\frac{1}{\det A}\mathrm{adj}(A)$$
   2. $$A^{-1}=\left[\begin{array}{cc}\frac{1}{2}& -\frac{1}{2}\\ -\frac{1}{2}& \frac{3}{2}\end{array}\right]$$
   3. $$A{A}^{-1}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$$
9. 9.

   Find the matrix inverse.

   $$A=\left[\begin{array}{cc}2& 5\\ 1& 3\end{array}\right],A^{-1}=?$$

   Answer: $\left[\begin{array}{cc}3& -5\\ -1& 2\end{array}\right]$

   Step-by-step solution

   1. $$A^{-1}=\frac{1}{\det A}\mathrm{adj}(A)$$
   2. $$A^{-1}=\left[\begin{array}{cc}3& -5\\ -1& 2\end{array}\right]$$
   3. $$A{A}^{-1}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$$
10. 10.

    Find the matrix inverse.

    $$A=\left[\begin{array}{cc}1& 3\\ 2& 7\end{array}\right],A^{-1}=?$$

    Answer: $\left[\begin{array}{cc}7& -3\\ -2& 1\end{array}\right]$

    Step-by-step solution

    1. $$A^{-1}=\frac{1}{\det A}\mathrm{adj}(A)$$
    2. $$A^{-1}=\left[\begin{array}{cc}7& -3\\ -2& 1\end{array}\right]$$
    3. $$A{A}^{-1}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$$
11. 11.

    Find the matrix inverse.

    $$A=\left[\begin{array}{cc}5& 2\\ 2& 1\end{array}\right],A^{-1}=?$$

    Answer: $\left[\begin{array}{cc}1& -2\\ -2& 5\end{array}\right]$

    Step-by-step solution

    1. $$A^{-1}=\frac{1}{\det A}\mathrm{adj}(A)$$
    2. $$A^{-1}=\left[\begin{array}{cc}1& -2\\ -2& 5\end{array}\right]$$
    3. $$A{A}^{-1}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$$
12. 12.

    Find the matrix inverse.

    $$A=\left[\begin{array}{cc}4& 3\\ 1& 1\end{array}\right],A^{-1}=?$$

    Answer: $\left[\begin{array}{cc}1& -3\\ -1& 4\end{array}\right]$

    Step-by-step solution

    1. $$A^{-1}=\frac{1}{\det A}\mathrm{adj}(A)$$
    2. $$A^{-1}=\left[\begin{array}{cc}1& -3\\ -1& 4\end{array}\right]$$
    3. $$A{A}^{-1}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$$
13. 13.

    Find the matrix inverse.

    $$A=\left[\begin{array}{cc}2& 1\\ 3& 2\end{array}\right],A^{-1}=?$$

    Answer: $\left[\begin{array}{cc}2& -1\\ -3& 2\end{array}\right]$

    Step-by-step solution

    1. $$A^{-1}=\frac{1}{\det A}\mathrm{adj}(A)$$
    2. $$A^{-1}=\left[\begin{array}{cc}2& -1\\ -3& 2\end{array}\right]$$
    3. $$A{A}^{-1}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$$
14. 14.

    Find the matrix inverse.

    $$A=\left[\begin{array}{cc}1& -1\\ 1& 1\end{array}\right],A^{-1}=?$$

    Answer: $\left[\begin{array}{cc}\frac{1}{2}& \frac{1}{2}\\ -\frac{1}{2}& \frac{1}{2}\end{array}\right]$

    Step-by-step solution

    1. $$A^{-1}=\frac{1}{\det A}\mathrm{adj}(A)$$
    2. $$A^{-1}=\left[\begin{array}{cc}\frac{1}{2}& \frac{1}{2}\\ -\frac{1}{2}& \frac{1}{2}\end{array}\right]$$
    3. $$A{A}^{-1}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$$
15. 15.

    Find the matrix inverse.

    $$A=\left[\begin{array}{cc}3& 2\\ 4& 3\end{array}\right],A^{-1}=?$$

    Answer: $\left[\begin{array}{cc}3& -2\\ -4& 3\end{array}\right]$

    Step-by-step solution

    1. $$A^{-1}=\frac{1}{\det A}\mathrm{adj}(A)$$
    2. $$A^{-1}=\left[\begin{array}{cc}3& -2\\ -4& 3\end{array}\right]$$
    3. $$A{A}^{-1}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$$
16. 16.

    Find the matrix inverse.

    $$A=\left[\begin{array}{cc}6& 1\\ 5& 1\end{array}\right],A^{-1}=?$$

    Answer: $\left[\begin{array}{cc}1& -1\\ -5& 6\end{array}\right]$

    Step-by-step solution

    1. $$A^{-1}=\frac{1}{\det A}\mathrm{adj}(A)$$
    2. $$A^{-1}=\left[\begin{array}{cc}1& -1\\ -5& 6\end{array}\right]$$
    3. $$A{A}^{-1}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$$
17. 17.

    Find the matrix inverse.

    $$A=\left[\begin{array}{cc}1& 4\\ 1& 5\end{array}\right],A^{-1}=?$$

    Answer: $\left[\begin{array}{cc}5& -4\\ -1& 1\end{array}\right]$

    Step-by-step solution

    1. $$A^{-1}=\frac{1}{\det A}\mathrm{adj}(A)$$
    2. $$A^{-1}=\left[\begin{array}{cc}5& -4\\ -1& 1\end{array}\right]$$
    3. $$A{A}^{-1}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$$
18. 18.

    Find the matrix inverse.

    $$A=\left[\begin{array}{cc}2& 7\\ 1& 4\end{array}\right],A^{-1}=?$$

    Answer: $\left[\begin{array}{cc}4& -7\\ -1& 2\end{array}\right]$

    Step-by-step solution

    1. $$A^{-1}=\frac{1}{\det A}\mathrm{adj}(A)$$
    2. $$A^{-1}=\left[\begin{array}{cc}4& -7\\ -1& 2\end{array}\right]$$
    3. $$A{A}^{-1}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$$

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