make video explaining the questions---Multiple Choice Questions
Choose the correct answer in each of the following:
1. If $A = [a_{ij}]_{m \times n}$ is a square matrix, then
(a) $m < n$.
(b) $m = n$.
(c) $m > n$.
(d) None of these.
2. Which of the given values of $x$ and $y$ make the following pair of matrices equal?
$\begin{bmatrix} 3x+7 & 5 \\ y+1 & 2-3x \end{bmatrix}$, $\begin{bmatrix} 0 & y-2 \\ 8 & 4 \end{bmatrix}$
(a) $x = -\frac{1}{3}, y = \frac{2}{3}$.
(b) $x = -\frac{2}{3}, y = 7$.
(c) Not possible to find.
(d) $x = -\frac{1}{3}, y = 7$.
3. If a matrix $A$ is both symmetric and skew symmetric, then $A$ is a [NCERT]
(a) scalar matrix.
(b) diagonal matrix.
(c) zero matrix.
(d) square matrix.
4. If $A, B$ are symmetric matrices of same order, then $(AB - BA)$ is a [NCERT]
(a) skew symmetric matrix.
(b) identity matrix.
(c) zero matrix.
(d) symmetric matrix.
5. If $A = \begin{bmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{bmatrix}$, then $A + A^T = I$, if the value of $\alpha$ is [NCERT]
(a) $\frac{3\pi}{2}$.
(b) $\pi$.
(c) $\frac{\pi}{6}$.
(d) $\frac{\pi}{3}$.
6. If $A = \begin{bmatrix} 5 & x \\ y & 0 \end{bmatrix}$ and $A = A^T$, then
(a) $x=0, y=5$.
(b) $x=y$.
(c) $x+y=5$.
(d) None of these.
7. Out of the given matrices, choose the matrix which is a scalar matrix:
(a) $\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$
(b) $\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$
(c) $\begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$
(d) $\begin{bmatrix} 0 & 0 \end{bmatrix}$
8. Matrices $A$ and $B$ will be inverse of each other only if [NCERT]
(a) $AB = BA$.
(b) $AB = BA = I$.
(c) $AB = BA = O$.
(d) $AB = O, BA = I$.
9. If $A$ is a square matrix such that $A^2 = A$, then $(I + A)^3 - 7A$ is equal to [NCERT]
(a) $I - A$.
(b) $A$.
(c) $I$.
(d) $3A$.
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Welcome to our matrix multiple choice quiz. We'll solve 9 questions step by step. Let's start with question 1. If A equals a i j subscript m by n is a square matrix, what can we say about m and n? A square matrix has equal number of rows and columns, so m equals n. The answer is option b.
Question 2 asks which values of x and y make two matrices equal. For matrices to be equal, corresponding elements must be equal. From the first element, 3x plus 7 equals 0, so x equals negative 7 thirds. From the fourth element, 2 minus 3x equals 4, so x equals negative 2 thirds. Since x cannot have two different values, it's not possible to find such values. The answer is option c.
Questions 3 and 4 deal with matrix properties. For question 3, if A is both symmetric and skew symmetric, then A transpose equals A and A transpose equals negative A. This means A equals negative A, so 2A equals zero matrix, therefore A is the zero matrix. Answer is c. For question 4, if A and B are symmetric, then AB minus BA transpose equals BA minus AB, which is negative of AB minus BA. This makes it skew symmetric. Answer is a.
Questions 5, 6, and 7 cover special matrices. For question 5, A plus A transpose gives a matrix with 2 cosine alpha on the diagonal. If this equals identity matrix, then 2 cosine alpha equals 1, so cosine alpha equals one half, which means alpha equals pi over 3. Answer is d. For question 6, if A equals A transpose, then x equals y. Answer is b. For question 7, a scalar matrix must be square with equal diagonal elements. Only option a is square. Answer is a.
Final two questions. Question 8 asks when matrices A and B are inverses. For matrices to be inverses, both AB and BA must equal the identity matrix I. Answer is b. Question 9 gives A squared equals A. We need to find I plus A cubed minus 7A. Since A cubed equals A, expanding I plus A cubed gives I plus 7A. Therefore I plus A cubed minus 7A equals I. Answer is c. We've completed all 9 matrix questions successfully!