Mathematics 10 — Exercise 3.1 · Q7

Full question

Find the unknown values from the equality of the matrices.

Step-by-step solution

1. These corresponding entries give a pair of linear equations.

   $$(p+q=6,p-2\times q=0)$$

2. Write the coefficients in A, the unknowns in X, and constants in B, giving AX = B.

   $$(\left[\begin{array}{cc}1& 1\\ 1& -2\end{array}\right],\left[\begin{array}{c}p\\ q\end{array}\right],\left[\begin{array}{c}6\\ 0\end{array}\right])$$

3. For a 2 × 2 determinant, multiply the main diagonal and subtract the other diagonal product.

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

4. Calculate the first diagonal product.

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

5. Calculate the second diagonal product.

   $$1\times 1=1$$

6. Subtract the second product, including its sign.

   $$D=-2-1$$

7. Simplify the subtraction.

   $$D=-3$$

8. For Dₓ replace the first coefficient column by the constants.

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

9. For a 2 × 2 determinant, multiply the main diagonal and subtract the other diagonal product.

   $$\mathrm{Dₓ}=6\times \left(-2\right)-1\times 0$$

10. Calculate the first diagonal product.

    $$6\times \left(-2\right)=-12$$

11. Calculate the second diagonal product.

    $$1\times 0=0$$

12. Subtract the second product, including its sign.

    $$\mathrm{Dₓ}=-12+0$$

13. Simplify the subtraction.

    $$\mathrm{Dₓ}=-12$$

14. For Dᵧ replace the second coefficient column by the constants.

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

15. For a 2 × 2 determinant, multiply the main diagonal and subtract the other diagonal product.

    $$\mathrm{Dᵧ}=1\times 0-6\times 1$$

16. Calculate the first diagonal product.

    $$1\times 0=0$$

17. Calculate the second diagonal product.

    $$6\times 1=6$$

18. Subtract the second product, including its sign.

    $$\mathrm{Dᵧ}=0-6$$

19. Simplify the subtraction.

    $$\mathrm{Dᵧ}=-6$$

20. Divide the replacement determinant by D.

    $$p=-\frac{12}{-3}$$

21. Reduce this quotient to obtain the variable value.

    $$p=4$$

22. Divide the replacement determinant by D.

    $$q=-\frac{6}{-3}$$

23. Reduce this quotient to obtain the variable value.

    $$q=2$$

24. Calculate this operation before combining the remaining terms.

    $$1\times 4=4$$

25. Calculate this operation before combining the remaining terms.

    $$1\times 2=2$$

26. Calculate this operation before combining the remaining terms.

    $$4+2=6$$

27. Check in this original equation by multiplying the coefficients by the calculated values.

    $$1\times 4+1\times 2=6$$

28. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 4=-4$$

29. Calculate this operation before combining the remaining terms.

    $$4-4=0$$

30. Check in this original equation by multiplying the coefficients by the calculated values.

    $$1\times 4-4=0$$

31. Check the values in the original two matrices.

    $$(\left[\begin{array}{cc}6& 5\\ 11& 0\end{array}\right],\left[\begin{array}{cc}6& 5\\ 11& 0\end{array}\right])$$

Answer

The final answer is shown below.