Matrix Products — Practice Problems and Worked Solutions

The formulas you need here

17 worked problems

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

1. 1.

   Multiply the matrices.

   $$\left[\begin{array}{cc}1& 2\\ 0& 1\end{array}\right]\left[\begin{array}{cc}2& 0\\ 3& 1\end{array}\right]$$

   Answer: $\left[\begin{array}{cc}8& 2\\ 3& 1\end{array}\right]$

   Step-by-step solution

   1. $$(AB{)}_{ij}={\text{row}}_i(A)\cdot {\text{column}}_j(B)$$
   2. $$AB=\left[\begin{array}{cc}8& 2\\ 3& 1\end{array}\right]$$
2. 2.

   Multiply the matrices.

   $$\left[\begin{array}{cc}2& -1\\ 1& 3\end{array}\right]\left[\begin{array}{cc}1& 2\\ 0& 1\end{array}\right]$$

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

   Step-by-step solution

   1. $$(AB{)}_{ij}={\text{row}}_i(A)\cdot {\text{column}}_j(B)$$
   2. $$AB=\left[\begin{array}{cc}2& 3\\ 1& 5\end{array}\right]$$
3. 3.

   Multiply the matrices.

   $$\left[\begin{array}{ccc}1& 0& 2\\ -1& 3& 1\end{array}\right]\left[\begin{array}{cc}2& 1\\ 0& -1\\ 1& 2\end{array}\right]$$

   Answer: $\left[\begin{array}{cc}4& 5\\ -1& -2\end{array}\right]$

   Step-by-step solution

   1. $$(AB{)}_{ij}={\text{row}}_i(A)\cdot {\text{column}}_j(B)$$
   2. $$AB=\left[\begin{array}{cc}4& 5\\ -1& -2\end{array}\right]$$
4. 4.

   Multiply the matrices.

   $$\left[\begin{array}{cc}3& 1\\ 2& -1\end{array}\right]\left[\begin{array}{cc}1& 4\\ 2& 0\end{array}\right]$$

   Answer: $\left[\begin{array}{cc}5& 12\\ 0& 8\end{array}\right]$

   Step-by-step solution

   1. $$(AB{)}_{ij}={\text{row}}_i(A)\cdot {\text{column}}_j(B)$$
   2. $$AB=\left[\begin{array}{cc}5& 12\\ 0& 8\end{array}\right]$$
5. 5.

   Multiply the matrices.

   $$\left[\begin{array}{cc}0& 2\\ -1& 3\end{array}\right]\left[\begin{array}{cc}4& -2\\ 1& 5\end{array}\right]$$

   Answer: $\left[\begin{array}{cc}2& 10\\ -1& 17\end{array}\right]$

   Step-by-step solution

   1. $$(AB{)}_{ij}={\text{row}}_i(A)\cdot {\text{column}}_j(B)$$
   2. $$AB=\left[\begin{array}{cc}2& 10\\ -1& 17\end{array}\right]$$
6. 6.

   Multiply the matrices.

   $$\left[\begin{array}{ccc}2& 1& -1\\ 0& 3& 2\end{array}\right]\left[\begin{array}{cc}1& 2\\ -1& 0\\ 2& 1\end{array}\right]$$

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

   Step-by-step solution

   1. $$(AB{)}_{ij}={\text{row}}_i(A)\cdot {\text{column}}_j(B)$$
   2. $$AB=\left[\begin{array}{cc}-1& 3\\ 1& 2\end{array}\right]$$
7. 7.

   Multiply the matrices.

   $$\left[\begin{array}{cc}1& 3\\ 0& 2\end{array}\right]\left[\begin{array}{cc}1& 0\\ 2& 1\end{array}\right]$$

   Answer: $\left[\begin{array}{cc}7& 3\\ 4& 2\end{array}\right]$

   Step-by-step solution

   1. $$(AB{)}_{ij}={\text{row}}_i(A)\cdot {\text{column}}_j(B)$$
   2. $$AB=\left[\begin{array}{cc}7& 3\\ 4& 2\end{array}\right]$$
8. 8.

   Multiply the matrices.

   $$\left[\begin{array}{cc}2& 0\\ 1& 4\end{array}\right]\left[\begin{array}{cc}3& 1\\ 0& 2\end{array}\right]$$

   Answer: $\left[\begin{array}{cc}6& 2\\ 3& 9\end{array}\right]$

   Step-by-step solution

   1. $$(AB{)}_{ij}={\text{row}}_i(A)\cdot {\text{column}}_j(B)$$
   2. $$AB=\left[\begin{array}{cc}6& 2\\ 3& 9\end{array}\right]$$
9. 9.

   Multiply the matrices.

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

   Answer: $\left[\begin{array}{cc}0& -5\\ 7& 6\end{array}\right]$

   Step-by-step solution

   1. $$(AB{)}_{ij}={\text{row}}_i(A)\cdot {\text{column}}_j(B)$$
   2. $$AB=\left[\begin{array}{cc}0& -5\\ 7& 6\end{array}\right]$$
10. 10.

    Multiply the matrices.

    $$\left[\begin{array}{cc}5& 1\\ 0& 2\end{array}\right]\left[\begin{array}{cc}1& 2\\ 3& 0\end{array}\right]$$

    Answer: $\left[\begin{array}{cc}8& 10\\ 6& 0\end{array}\right]$

    Step-by-step solution

    1. $$(AB{)}_{ij}={\text{row}}_i(A)\cdot {\text{column}}_j(B)$$
    2. $$AB=\left[\begin{array}{cc}8& 10\\ 6& 0\end{array}\right]$$
11. 11.

    Multiply the matrices.

    $$\left[\begin{array}{ccc}1& 0& 3\\ 2& 1& 0\end{array}\right]\left[\begin{array}{cc}1& 1\\ 0& 2\\ 2& 0\end{array}\right]$$

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

    Step-by-step solution

    1. $$(AB{)}_{ij}={\text{row}}_i(A)\cdot {\text{column}}_j(B)$$
    2. $$AB=\left[\begin{array}{cc}7& 1\\ 2& 4\end{array}\right]$$
12. 12.

    Multiply the matrices.

    $$\left[\begin{array}{cc}3& 2\\ 1& 0\end{array}\right]\left[\begin{array}{cc}0& 1\\ 2& 3\end{array}\right]$$

    Answer: $\left[\begin{array}{cc}4& 9\\ 0& 1\end{array}\right]$

    Step-by-step solution

    1. $$(AB{)}_{ij}={\text{row}}_i(A)\cdot {\text{column}}_j(B)$$
    2. $$AB=\left[\begin{array}{cc}4& 9\\ 0& 1\end{array}\right]$$
13. 13.

    Multiply the matrices.

    $$\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]\left[\begin{array}{cc}2& 3\\ 4& 5\end{array}\right]$$

    Answer: $\left[\begin{array}{cc}6& 8\\ 6& 8\end{array}\right]$

    Step-by-step solution

    1. $$(AB{)}_{ij}={\text{row}}_i(A)\cdot {\text{column}}_j(B)$$
    2. $$AB=\left[\begin{array}{cc}6& 8\\ 6& 8\end{array}\right]$$
14. 14.

    Multiply the matrices.

    $$\left[\begin{array}{cc}4& -1\\ 2& 3\end{array}\right]\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$$

    Answer: $\left[\begin{array}{cc}4& -1\\ 2& 3\end{array}\right]$

    Step-by-step solution

    1. $$(AB{)}_{ij}={\text{row}}_i(A)\cdot {\text{column}}_j(B)$$
    2. $$AB=\left[\begin{array}{cc}4& -1\\ 2& 3\end{array}\right]$$
15. 15.

    Multiply the matrices.

    $$\left[\begin{array}{ccc}2& 3& 1\\ 1& 0& 2\end{array}\right]\left[\begin{array}{cc}1& 2\\ 0& 1\\ 3& 0\end{array}\right]$$

    Answer: $\left[\begin{array}{cc}5& 7\\ 7& 2\end{array}\right]$

    Step-by-step solution

    1. $$(AB{)}_{ij}={\text{row}}_i(A)\cdot {\text{column}}_j(B)$$
    2. $$AB=\left[\begin{array}{cc}5& 7\\ 7& 2\end{array}\right]$$
16. 16.

    Multiply the matrices.

    $$\left[\begin{array}{cc}0& 5\\ 2& 1\end{array}\right]\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$$

    Answer: $\left[\begin{array}{cc}15& 20\\ 5& 8\end{array}\right]$

    Step-by-step solution

    1. $$(AB{)}_{ij}={\text{row}}_i(A)\cdot {\text{column}}_j(B)$$
    2. $$AB=\left[\begin{array}{cc}15& 20\\ 5& 8\end{array}\right]$$
17. 17.

    Multiply the matrices.

    $$\left[\begin{array}{cc}6& 2\\ 1& 3\end{array}\right]\left[\begin{array}{cc}2& 0\\ 1& 1\end{array}\right]$$

    Answer: $\left[\begin{array}{cc}14& 2\\ 5& 3\end{array}\right]$

    Step-by-step solution

    1. $$(AB{)}_{ij}={\text{row}}_i(A)\cdot {\text{column}}_j(B)$$
    2. $$AB=\left[\begin{array}{cc}14& 2\\ 5& 3\end{array}\right]$$

More from Linear Algebra

• Vector Operations
• Dot Products
• Linear Systems
• Row Reduction
• Determinants
• Matrix Inverses
• Linear Independence
• Rank & Basis
• Orthogonal Projections
• Linear Transformations
• Eigenvalues