Matrices

If A and B are square matrices of the same order and B is a skew-symmetric matrix, then we need to determine the nature of .

Since B is a skew-symmetric matrix, we have .

Let . To determine if C is symmetric or skew-symmetric, we need to find .

Using the property , we have:

Since B is skew-symmetric, , so:

Since , C is a skew-symmetric matrix.

Therefore, the correct option is (D) Skew symmetric matrix.

Given , we need to find .

First, let's find :

Now, let's find :

Thus, .

Therefore, the correct option is (C) I.

Given equation:

$ A \begin{pmatrix} 2 & 1 \\ 3 & 2 \end{pmatrix} \begin{pmatrix} 2 & 1 \\ 3 & 2 \end{pmatrix} \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} 2 & 1 \\ 3 & 2 \end{pmatrix} a = 2 b = 1 c = 3 d = 2 ad - bc = (2)(2) - (1)(3) = 4 - 3 = 1 A \begin{pmatrix} 2 & -1 \\ -3 & 2 \end{pmatrix} $.

Let be a 2 × 2 symmetric matrix with integer entries. A symmetric matrix has the form , where a, b, and c are integers.

A matrix is invertible if its determinant is non-zero. The determinant of is given by .

We need to find the condition under which is invertible, i.e., .

1. The first column of M is the transpose of the second row of M. This means , so and . Therefore, . In this case, , so is not invertible.
2. The second row of M is the transpose of the first column of M. This statement simply defines what a symmetric matrix is. It doesn't tell us anything more about the relationship between a, b and c. So , can be 0 or non-zero, hence we don't have sufficient information
3. M is a diagonal matrix with non-zero entries in the principal diagonal. A diagonal matrix has the form , where and . In this case, , and . Since and , , so is invertible.
4. The product of entries in the principal diagonal of M is the product of entries in the other diagonal. This means . Therefore, , so is not invertible.

Therefore, is invertible if is a diagonal matrix with non-zero entries in the principal diagonal.

First, let's calculate the product : The product of two matrices is calculated as follows: Now, we multiply the rows of with the columns of : - First row, first column:
First row, second column:
Second row, first column:
Second row, second column:
Therefore, .
Now, we need to find , which is the transpose of .
The transpose of a matrix is obtained by switching its rows and columns:

Thus, the correct answer is , which corresponds to option (B).

We are given: Note that: So, since , the matrix is nilpotent of index 2. Now, expand using the binomial theorem: But since , all terms with vanish. So, we keep only terms for and : Correct answer:

Given:
Set X has 7 elements and Set Y has 8 elements.

To find: The number of bijections from X to Y.

Concept:
A bijection is a function that is both one-to-one (injective) and onto (surjective).
For a bijection to exist between two sets, they must have the same number of elements.
Since |X| = 7 and |Y| = 8, and 7 ≠ 8, it is not possible to form a bijection between X and Y.

Answer: 0