Mathematics 10 — Exercise 6.2 · Q5

Full question

Find the components of a+5b.

Step-by-step solution

1. Multiply every matrix entry by the scalar.

   $$(5,\left[\begin{array}{c}3\\ -5\end{array}\right])$$

2. Entry (1,1): multiply its value by the scalar.

   $$5\times 3=15$$

3. Entry (2,1): multiply its value by the scalar.

   $$5\times \left(-5\right)=-25$$

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

   $$\left[\begin{array}{c}15\\ -25\end{array}\right]$$

5. The orders match. Combine only entries in the same row and column.

   $$(\left[\begin{array}{c}-2\\ 7\end{array}\right],\left[\begin{array}{c}15\\ -25\end{array}\right])$$

6. Entry (1,1): add the corresponding entries.

   $$-2+15=13$$

7. Entry (2,1): add the corresponding entries.

   $$7-25=-18$$

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

   $$\left[\begin{array}{c}13\\ -18\end{array}\right]$$

9. The first component multiplies î, the second multiplies ĵ.

   $$13\times î-18\times ĵ$$

Answer

The final answer is shown below.