Unit 3: Matrices and Determinants — Online MCQ Test
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Practice Unit 3: Matrices and Determinants with a free chapter-wise online MCQ test.
This chapter covers: Matrices algebra of matrices type of matrices determinants and matrices of order two and three evaluation of determinants area of triangles usingdeterminants; Adjoint and inverse o....
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Unit 3: Matrices and Determinants — Important Questions & Answers
For the matrices A = [[3, 4], [-1, 1]] and B = [[29, 49], [-13, 18]], if A^15 + B = [[x, 0], [0, y]], then which one is true?
- A. x = 5, y = 7
- B. x = 18, y = 11
- C. x = 11, y = 2
- D. x = 16, y = 3
Answer: C. x = 11, y = 2
By calculating powers of A, identify the pattern for A^n. A^15 is found and added to B. Comparing the result matrix [[x, 0], [0, y]], we find x = 11 and y = 2.
By calculating powers of A, identify the pattern for A^n. A^15 is found and added to B. Comparing the result matrix [[x, 0], [0, y]], we find x = 11 and y = 2.
For the matrices A = [[3, -4], [1, -1]] and B = [[29, -49], [13, -18]], if A^n + B = [[x, 0], [y, 0]], then among the following which one is true?
- A. x = 5, y = 7
- B. x = 18, y = 11
- C. x = 11, y = 2
- D. x = 16, y = 3
Answer: C. x = 11, y = 2
Calculating powers of A, A^2 = [[5, -8], [2, -3]], A^n = [[2n+1, -4n], [n, 1-2n]]. Using A^n + B = [[11, 0], [2, 0]], we get x=11 and y=2.
Calculating powers of A, A^2 = [[5, -8], [2, -3]], A^n = [[2n+1, -4n], [n, 1-2n]]. Using A^n + B = [[11, 0], [2, 0]], we get x=11 and y=2.
If the system of equations 3x + y + 4z = 3, 2x + alpha y + z = -2, x + 2y + z = 4 has no solution, then the value of alpha is equal to:
- A. 19
- B. 4
- C. 13
- D. 23
Answer: A. 19
For no solution, the determinant of the coefficient matrix must be zero. Solving det(A) = 0 gives alpha = 19.
For no solution, the determinant of the coefficient matrix must be zero. Solving det(A) = 0 gives alpha = 19.
If [x, y, z]^T is a solution of the system of equations AX = B, where adj(A) = [[4, 2, 2], [5, 0, 5], [1, 2, 3]] and B = [4, 0, 2]^T, then x + y + z is equal to:
- A. 3
- B. 3/2
- C. 1
- D. 2
Answer: D. 2
Using the relation X = (1/|A|) * adj(A) * B. Calculate the determinant |adj(A)| = 4(0-10) - 2(15-5) + 2(10-0) = -40 - 20 + 20 = -40. Then |A|^2 = |-40| = 40, so |A| = +/- sqrt(40). After solving for the vector X, summing the components x + y + z results in 2.
Using the relation X = (1/|A|) * adj(A) * B. Calculate the determinant |adj(A)| = 4(0-10) - 2(15-5) + 2(10-0) = -40 - 20 + 20 = -40. Then |A|^2 = |-40| = 40, so |A| = +/- sqrt(40). After solving for the vector X, summing the components x + y + z results in 2.
The number of symmetric matrices of order $3$, with all the entries from the set $\{0,1,2,3,4,5,6,7,8,9\}$, is
- A. $10^9$
- B. $9^{10}$
- C. $10^6$
- D. $6^{10}$
Answer: C. $10^6$
A symmetric $3\times 3$ matrix is determined by the $3$ diagonal entries and $3$ entries above the diagonal: \[ \begin{pmatrix} a&b&c\\b&d&e\\c&e&f \end{pmatrix}. \] There are $6$ independent entries, and each can be chosen in $10$ ways from $\{0,1,\dots,9\}$. So the number of such matrices is \[ 10^6. \] Thus the correct option is $c$.
A symmetric $3\times 3$ matrix is determined by the $3$ diagonal entries and $3$ entries above the diagonal: \[ \begin{pmatrix} a&b&c\\b&d&e\\c&e&f \end{pmatrix}. \] There are $6$ independent entries, and each can be chosen in $10$ ways from $\{0,1,\dots,9\}$. So the number of such matrices is \[ 10^6. \] Thus the correct option is $c$.