P2 Matrices Test
- 1. ppr maths nbk SPM FORMAT QUESTIONS PAPER 2 ⎛3 − 2⎞ ⎛1 0⎞ 1. M is a 2 x 2 matrix where M ⎜ ⎜ ⎟ =⎜ ⎟ ⎜ ⎟. ⎟ ⎝5 − 4⎠ ⎝ 0 1⎠ (a) Find the matrix M (b) Write the following simultaneous linear equations as matrix equation 3x – 2y = 7 5x – 4y = 9 Hence, calculate the values of x and y using matrices. ⎛3 − 4⎞ ⎛− 6 p⎞ 2. (a) The inverse matrix of ⎜ ⎜ 5 − 6 ⎟ is m ⎟ ⎜ ⎜− 5 ⎟. ⎝ ⎠ ⎝ 3⎟ ⎠ Find the value of m and p. (b) Using matrices, calculate the value of x and of y that satisfy the following simultaneous linear equations: 3x – 4y = -1 5x – 6y = 2 ⎛ 2 − 5⎞ ⎛ 3 h⎞ 3. It is given that matrix P = ⎜⎜ 1 3 ⎟ and matrix Q = k ⎜ − 1 2 ⎟ such that ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎛1 0⎞ PQ = ⎜ ⎜0 1⎟ . ⎟ ⎝ ⎠ (a) Find the value of k and h. (b) Using matrices, calculate the value of x and of y that satisfy the following simultaneous linear equations: 2x – 5y = -17 x + 3y = 8
- 2. ppr maths nbk ⎛2 1 ⎞ ⎛ − 4 − 1⎞ 4. (a) The inverse matrix of ⎜ ⎜ 3 − 4 ⎟ is m ⎟ ⎜ ⎜ p ⎟. ⎝ ⎠ ⎝ 2⎟⎠ Find the value of m and p. (b) Using matrices, calculate the value of x and y that satisfy the following simultaneous linear equations: 2x + y = 4 3x – 4y = 17. 1⎛ 4 1 ⎞ ⎛ − 2 m⎞ ⎛1 0⎞ 5. It is given that ⎜ ⎜ − 6 − 2⎟ ⎜ 6 4 ⎟ = ⎜0 1⎟ . ⎟⎜ ⎟ ⎜ ⎟ k ⎝ ⎠⎝ ⎠ ⎝ ⎠ (a) Find the value of k. (b) Find the value of m. (c) Hence, using matrices, calculate the value of v and w that satisfy the following matrix equation: ⎛ 4 1 ⎞ ⎛v⎞ ⎛ 8 ⎞ ⎜ ⎟⎜ ⎟ = ⎜ ⎜ − 6 − 2⎟ ⎜ w⎟ ⎜ − 10 ⎟ ⎟ ⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎛ a 3⎞ 6. Given matrix N = ⎜ ⎜ 6 9⎟ . ⎟ ⎝ ⎠ (a) If the determinant for matrix N is zero, find the value of a. (b) If a = 1, (i) find the inverse of matrix N, (ii) using matrix method, find the values of h and k that satisfy the following matrix equation : ⎛ 1 3⎞ ⎛ h ⎞ ⎛ − 5⎞ ⎜ ⎜ 6 9⎟ ⎜ k ⎟ = ⎜ 6 ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
- 3. ppr maths nbk ⎛ 6 3⎞ 1 ⎛− 3 3 ⎞ 7. If A is the matrix ⎜ ⎜ a b ⎟ and the inverse matrix of A is a ⎜ a − 6 ⎟ , find the ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ values of a and b. Hence , using matrices, calculate the values of x and y that satisfy the following simultaneous linear equation: 6x + 3y = 3 Ax + by = 5 ⎛5 r ⎞ 8. Given matrix G = ⎜ ⎜ 4 − 2⎟ , ⎟ ⎝ ⎠ (a) find the value of r if G does not have an inverse matrix, (b) find the inverse matrix of G, if r = -2, (c) calculate by using matrices, the values of v and w that satisfy the following matrix equation : ⎛ 5 − 2⎞ ⎛ v ⎞ ⎛1⎞ ⎜ ⎜ 4 − 2⎟ ⎜ w⎟ = ⎜ 2⎟ ⎟⎜ ⎟ ⎜ ⎟ ⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎛ 5 − 3⎞ ⎛ − 2 3⎞ 9. (a)Given that the inverse matrix of ⎜ ⎜ 4 − 2 ⎟ is m ⎜ p 5 ⎟ , find the values of m ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ and p. (b) Using matrices, find the values of x and y that satisfy the following simultaneous equations. 5x – 3y = 1 4x – 2y = 2 ⎛ 4 5⎞ ⎛1 0⎞ 10. Given matrix P = ⎜ ⎜ 6 8 ⎟ and matrix PQ = ⎟ ⎜ ⎜0 1⎟ ⎟ ⎝ ⎠ ⎝ ⎠ (a) find matrix Q, and (b) hence, calculate by using matrix method, the values of m and n that satisfy the following simultaneous linear equations: 4m + 5n = 7 6m + 8n = 1
- 4. ppr maths nbk ANSWERS PAPER 2 1 ⎛ − 4 2⎞ 1. (a) M=- ⎜ ⎟ 2 ⎜ − 5 3⎟ ⎝ ⎠ ⎛3 − 2⎞ ⎛ x ⎞ ⎛ 7⎞ (b) ⎜ ⎜5 − 4⎟ ⎜ y ⎟ = ⎜ 9⎟ ⎟⎜ ⎟ ⎜ ⎟ ⎝ ⎠⎝ ⎠ ⎝ ⎠ x=5 y = -4 1 2. (a) m= 2 p=4 (b) x=7 11 y= 2 1 3. (a) k= 11 h=5 (b) x = -1 y=3 1 4. (a) m=- 11 p = -3 (b) x=3 y=-2 5. (a) k = -2 m = -1 (b) v=3 w = -4 6. (a) 2
- 5. ppr maths nbk ⎛ 1 ⎞ ⎜−1 ⎟ (c) (i) ⎜ 3 ⎟ ⎜ 2 − 1⎟ ⎜ ⎟ ⎝3 9⎠ (iii) h = 7, k = -4 2 1 7. a= 9, b = 3, x= ,y= - 3 3 5 8. r=- 2 ⎛ 1 − 1⎞ (b) ⎜ 5⎟ ⎜2 ⎟ ⎝ 2⎠ (c) v =-1, w = − 3 1 9. (a) m = p = -4 2 (b) x = 2, y = 3 ⎛ 5⎞ 10. (a) ⎜ 4 − ⎟ ⎜ 2⎟ ⎝ −3 2 ⎠ (b) m = 3, n = -1