Mathematics 10 — Exercise 3.4 · Q1 (i) AB

Full question

Find AB, if possible.

Step-by-step solution

1. 2 columns in the first matrix match 2 rows in the second. The product has 2 rows and 2 columns.

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

2. Entry (1,1): pair row 1 with column 1 and add their products.

   $$1\times 3+2\times 1$$

3. Calculate this individual product, including its signs.

   $$1\times 3=3$$

4. Calculate this individual product, including its signs.

   $$2\times 1=2$$

5. Add the calculated products to obtain this entry.

   $$3+2=5$$

6. Entry (1,2): pair row 1 with column 2 and add their products.

   $$1\times 2+2\times \left(-1\right)$$

7. Calculate this individual product, including its signs.

   $$1\times 2=2$$

8. Calculate this individual product, including its signs.

   $$2\times \left(-1\right)=-2$$

9. Add the calculated products to obtain this entry.

   $$2-2=0$$

10. Entry (2,1): pair row 2 with column 1 and add their products.

    $$-3+0\times 1$$

11. Calculate this individual product, including its signs.

    $$\left(-1\right)\times 3=-3$$

12. Calculate this individual product, including its signs.

    $$0\times 1=0$$

13. Add the calculated products to obtain this entry.

    $$-3+0=-3$$

14. Entry (2,2): pair row 2 with column 2 and add their products.

    $$-2+0\times \left(-1\right)$$

15. Calculate this individual product, including its signs.

    $$\left(-1\right)\times 2=-2$$

16. Calculate this individual product, including its signs.

    $$0\times \left(-1\right)=0$$

17. Add the calculated products to obtain this entry.

    $$-2+0=-2$$

18. Place the calculated entries in their original row and column positions.

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

Answer

The final answer is shown below.