Worksheet on Matrix

In Worksheet on matrix the questions are based on finding unknown elements and matrices from matrix equation.

1. Let A = \(\begin{bmatrix} 1 & 2\\ 2 & 3 \end{bmatrix}\), B = \(\begin{bmatrix} 2 & 1\\ 3 & 2 \end{bmatrix}\) and C =  \(\begin{bmatrix} 1 & 3\\ 3 & 1 \end{bmatrix}\)

(i) Find the matrix C(B – A).

(ii) Find A(B + C).

(iii) Prove that A(B + C) = AB + AC.

2. If A =  \(\begin{bmatrix} 3 & 2\\ 1 & 0 \end{bmatrix}\), B =  \(\begin{bmatrix} -1 & 3\\ 0 & 1 \end{bmatrix}\) then verify the truth of the following.

(i) (A + B)2 = A2 + B2 + 2AB

(ii) (A + B)(A – B) = A2 – B2


3. If X =  \(\begin{bmatrix} 4 & 1\\ -1 & 2 \end{bmatrix}\), show that 6X – X2 = 9I, where I is the unit matrix.

4. (i) Show that X = \(\begin{bmatrix} 1 & 2\\ 2 & 1 \end{bmatrix}\) satisfies the relation X2 – 2X – 3I = O, where I is the unit matrix of order 2 × 2 and O is the null matrix of order 2 × 2.

(ii) Let A = \(\begin{bmatrix} 1 & 0\\ 2 & 1 \end{bmatrix}\), B = \(\begin{bmatrix} 2 & 3\\ -1 & 0 \end{bmatrix}\). Find A2 + AB + B2.


5. Find the matrix X from the matrix equation \(\begin{bmatrix} 2 & 1\\ 5 & 0 \end{bmatrix}\) – 3X = \(\begin{bmatrix} -7 & 4\\ 2 & 6 \end{bmatrix}\).

6. (i) If A = \(\begin{bmatrix} 1 & x\\ 0 & 1 \end{bmatrix}\) and \(\begin{bmatrix} 3 & 4\\ 5 & y \end{bmatrix}\) such that A + 2B = 5\(\begin{bmatrix} \frac{7}{5} & 0\\ 2 & 1 \end{bmatrix}\) then find x and y.

(ii) If 2\(\begin{bmatrix} 3 & 4\\ 5 & x \end{bmatrix}\) + \(\begin{bmatrix} 1 & y\\ 0 & 1 \end{bmatrix}\) = \(\begin{bmatrix} 7 & x\\ 10 & 5 \end{bmatrix}\), find x and y.


7. If \(\begin{bmatrix} 4\\ x\\ -y \end{bmatrix}\)\(\begin{bmatrix} 2 & 1 \end{bmatrix}\) = \(\begin{bmatrix} 8 & 4\\ 6 & 3\\ -8 & -4 \end{bmatrix}\), find x and y.

8. If \(\begin{bmatrix} 4 & -5\\ 6 & 7 \end{bmatrix}\)\(\begin{bmatrix} x\\ y \end{bmatrix}\) = \(\begin{bmatrix} -1\\ 2 \end{bmatrix}\), find x and y.

9. If \(\begin{bmatrix} 1 & 2\\ 3 & 3 \end{bmatrix}\)\(\begin{bmatrix} x & 0\\ 0 & y \end{bmatrix}\) = \(\begin{bmatrix} x & 0\\ 9 & 0 \end{bmatrix}\), find x and y.

10. If \(\begin{bmatrix} -3 & 2\\ 0 & -5 \end{bmatrix}\)\(\begin{bmatrix} x\\ 2 \end{bmatrix}\) = \(\begin{bmatrix} -5\\ y \end{bmatrix}\), find x and y.

11. If \(\begin{bmatrix} 2 & 3\\ -4 & 0 \end{bmatrix}\)\(\begin{bmatrix} x & 1\\ y & 1 \end{bmatrix}\) = \(\begin{bmatrix} 4 & z\\ -3 & -4 \end{bmatrix}\), then find x, y and z.

Worksheet on Matrix

12. Let A = \(\begin{bmatrix} 2 & 12\\ 0 & 1 \end{bmatrix}\) and B = \(\begin{bmatrix} 4 & x\\ 0 & 1 \end{bmatrix}\). If A2 = B, find x.

13. Let A = \(\begin{bmatrix} 1 & 3\\ 1 & 3 \end{bmatrix}\) and B = \(\begin{bmatrix} 3 & a\\ b & 2 \end{bmatrix}\). If AB = O, where O is the null matrix, find a and b.

14. Find x and y if \(\begin{bmatrix} x & 3x\\ y & 4y \end{bmatrix}\)\(\begin{bmatrix} 2\\ 1 \end{bmatrix}\) = \(\begin{bmatrix} 5\\ 12 \end{bmatrix}\).

15. Let A = \(\begin{bmatrix} 2 & 1\\ 3 & 4 \end{bmatrix}\). Find the matrix B such that A2 = 2A + 3B.

16. Let A = \(\begin{bmatrix} 1 & -3\\ 2 & 4 \end{bmatrix}\), B = \(\begin{bmatrix} -1\\ 2 \end{bmatrix}\) and C is a matrix such that AC = B then find the matrix C.

17. Let A = \(\begin{bmatrix} 2 & 1\\ 3 & 4 \end{bmatrix}\). Find the matrix B such that AB = I, where I is the unit matrix of the order 2 × 2.

18. Given A = \(\begin{bmatrix} 2 & -1\\ 2 & 0 \end{bmatrix}\), B = \(\begin{bmatrix} -3 & 2\\ 4 & 0 \end{bmatrix}\) and C = \(\begin{bmatrix} 1 & 0\\ 0 & 2 \end{bmatrix}\). Find the matrix X such that A + X = 2B + C.

19. If \(\begin{bmatrix} 1 & 4\\ -2 & 3 \end{bmatrix}\) + 2M = 3\(\begin{bmatrix} 3 & 2\\ 0 & -3 \end{bmatrix}\), find the matrix M.

20. (i) Show that X = \(\begin{bmatrix} 1 & 2\\ 2 & 1 \end{bmatrix}\) satisfies the relation X2 – 2X – 3I = O, where I is the unit matrix of order 2 × 2 and O is the null matrix of order 2 × 2.

(ii) Let A = \(\begin{bmatrix} 1 & 0\\ 2 & 1 \end{bmatrix}\), B = \(\begin{bmatrix} 2 & 3\\ -1 & 0 \end{bmatrix}\). Find A2 + AB + B2.

Answers:

 

1. (i) \(\begin{bmatrix} 4 & -4\\ 4 & -4 \end{bmatrix}\)

(ii) \(\begin{bmatrix} 15 & 10\\ 24 & 17 \end{bmatrix}\)


2. (i) False

(ii) False


4. (ii) \(\begin{bmatrix} 4 & 9\\ 5 & 4 \end{bmatrix}\)

5. \(\begin{bmatrix} 3 & -1\\ 1 & -2 \end{bmatrix}\)

6. (i) x = -8, y = 2

(ii) x = 2, y = -8


7. x = 3, y = -4

8. x = \(\frac{3}{58}\), y = \(\frac{7}{29}\)

9. x = 3, y = 0

10. x = 3, y = -10

11. x = \(\frac{3}{4}\), y = \(\frac{5}{6}\), z = 5

12. x = 36

13. a = -6, b = -1

14. x = 1, y = 2

15. \(\begin{bmatrix} 1 & \frac{4}{3}\\ 4 & \frac{11}{3} \end{bmatrix}\)

16. \(\begin{bmatrix} \frac{1}{5}\\ \frac{2}{5} \end{bmatrix}\)

17. \(\begin{bmatrix} \frac{4}{5} & -\frac{1}{5}\\ -\frac{3}{5} & \frac{2}{5} \end{bmatrix}\)

18. \(\begin{bmatrix} -7 & 5\\ 6 & 2 \end{bmatrix}\)

19. \(\begin{bmatrix} 4 & 1\\ 1 & -6 \end{bmatrix}\).

20. (ii) \(\begin{bmatrix} 4 & 9\\ 5 & 4 \end{bmatrix}\).





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