Matrices

• From the top-left element: gives
• Check with the bottom-right element: also gives

1. We are considering matrices of the form: $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ where each element is from the set {1, -1, 0, 0}.
2. First, select 2 positions out of the 4 for the 0s. The number of ways to choose 2 positions out of 4 is given by $\genfrac{}{}{0ex}{}{\left(4\right)}{}$. Thus, the number of ways is: $4^2=6$.
3. Now that the remaining 2 positions must be filled with 1 and -1, there are 2 ways to arrange them in the selected positions.
4. Thus, the total number of matrices is obtained by multiplying these outcomes:

Given that A is a square matrix of order such that (meaning A is an idempotent matrix) and I is the identity matrix of the same order, we need to find the value of .

First, note the following relations:

• , so we can substitute wherever appears.
• is also a matrix of the same order.

To solve , we expand as follows:

.

Expanding further:

• .
• Since , the expression becomes .
• The simplification gives .

Continuing with , we apply the expansion again:

.

Thus, .

Substituting back into the main expression:

.

Therefore, the value of the expression is .

First, we clarify and correct the matrix notation given in the question. We have matrices and as:

To find , we multiply matrices and :

The resulting matrix is obtained by computing each element through matrix multiplication:

1.

2.

3.

4.

5. Therefore, the matrix is:

We are required to find :

It seems the correct matrix multiplication was entered incorrectly. Therefore, let's correctly calculate:

Upon solving correctly, :

The correct sum should be .

This resolves the given question correctly, with the calculated option matching as .

To find the product , where and , we will perform matrix multiplication.
The resulting matrix is computed as follows:

For the element :

For the element :

For the element :

For the element :

For the element :

For the element :

For the element :

For the element :

For the element :

Hence, the product matrix is:

1. For the first element:
   
   Solve for :
   
2. For the second element:
   
   Substitute :
   
3. Confirming from the third element:
   
   Substitute and check with :
   
4. Finally, for the fourth element:
   
5. Calculate with and :
   

Given the matrix . For the matrix to be symmetric, the elements across the main diagonal must be equal, i.e., the matrix should satisfy the condition . This implies: for all and .

The matrix is .

For symmetry, equate the off-diagonal elements: .

To find the value of , solve the equation:

Subtract from both sides:

Subtract 6 from both sides:

Divide by 3:

Thus, when , the matrix becomes , which is symmetric. Therefore, the value of is 2.

Given and , we need to find the condition for . Calculating :

For :

This implies:

Rearranging gives:

To solve for the inverse of the matrix , given that it satisfies the equation , where is the identity matrix of order 2, follow these steps:

1. Understand that is equivalent to the matrix polynomial .

2. Calculate where represents the eigenvalues of . The characteristic equation is .

3. Find the determinant: .

4. Simplify: , indicating eigenvalues satisfying the identity polynomial equation.

5. The inverse relation from the polynomial is .

6. Calculate .

7. Now use the inverse calculation: .

Thus, the correct answer is .

To solve for in the equation , we start by evaluating the properties of the given matrix .

Matrix is given by:

Step 1: Transpose of Matrix

The transpose of a diagonal matrix is simply the matrix itself, since transposing changes component (i, j) to (j, i), both being equal for diagonal elements. Thus:

Step 2: Calculating

Since , we have:

To compute , we first need :

Then, :

Step 3: Solve for in

We equate and :

This gives two scalar equations:

Simplifying, we get:

Solving for , we divide both sides by :

Thus, the value of is .

The given matrix is: . A matrix is skew-symmetric if . This means each element satisfies the condition .

For element in position and :
. This condition is satisfied.

For position and :
.

For position and :
.

Now, calculate :
.

Thus, the value of is 10.

If two matrices are equal, then their corresponding elements must be equal. Therefore, we have the following equations from the given matrices:

From equation (1):

From equation (3):

Now, calculate :

Thus, the value of is .

To solve for the value of , given that , , and , we start by calculating the product matrix .

Compute :

Equate this with the given product matrix:

Comparing elements, we have:

(since )

Substitute into :

Verify with the second equation:

which holds true.

Now, calculate :

,

Thus, the value of is 9.

To solve the problem, we must first understand the properties of the given matrix . Let's analyze the matrix according to the options provided:

Square Matrix: A square matrix is one that has the same number of rows and columns. Matrix is , thus, it is a square matrix. Therefore, option (A) is correct.

Diagonal Matrix: A diagonal matrix has all off-diagonal elements equal to zero. In matrix , the off-diagonal elements are , which are not zero unless . Therefore, matrix is not necessarily a diagonal matrix unless these conditions are true, so option (B) is not correct in general.

Symmetric Matrix: A symmetric matrix satisfies (i.e., for all i, j). In matrix , but , which are not equal, confirming . Therefore, matrix is not symmetric, making option (C) incorrect.

Skew-Symmetric Matrix: A skew-symmetric matrix satisfies (i.e., for all i, j). In matrix , and , and , and , which are all consistent with the definition of skew-symmetry. Therefore, matrix is skew-symmetric, making option (D) correct.

Given the analysis, the correct options are (A) and (D), corresponding to the answer choice: (A) and (D) only

To solve for , we first need to find the product and then take its transpose. Given:

and .

The product is given by:

.

Calculating each entry:

• First row, first column: .
• First row, second column: .
• Second row, first column: .
• Second row, second column: .
• Third row, first column: .
• Third row, second column: .

Hence,

.

The transpose of a matrix is obtained by swapping its rows with columns. Thus, the transpose is:

.

This corresponds to the option:

Given matrices:


The transpose of gives us matrix :

To find matrix , we perform matrix multiplication:

The element at position (1,1) is calculated as follows:

The element at position (1,2) is calculated as follows:

The element at position (2,1) is calculated as follows:

The element at position (2,2) is calculated as follows:

Thus, matrix is:

Therefore, the correct answer is:

To solve for given and , let's explore the properties of matrices and alongside the identity matrix . The conditions state:

First, let's find :

If we left multiply both sides of by , we have:

• Since , it simplifies to .

Next, consider the expression :

• Substituting into the expression, we get:

Thus, the expression simplifies to: 3Y

So, the correct option is 3Y.

To find , we need to first find the inverses of matrices and . The inverse of a 2x2 matrix is given by: , provided that .

Step 1: Find

Matrix .

The determinant of is:

.

Step 2: Find

Matrix .

The determinant of is:

.

Step 3: Calculate

Now multiply and :

Using matrix multiplication:

This simplifies to: .

The correct answer is: .

To determine the domain of the function , we must first understand the constraints of the inverse cosine function. The inverse cosine function, , is defined only for values of in the interval .

For to be valid, the expression must also lie within the domain of the inverse cosine function: .

Let us solve the inequalities:

1.

Divide both sides by 7:

2.

Again, divide both sides by 7:

Combining these two inequalities, we find:

Therefore, the domain of the function is .

Given the matrix equation , it indicates that is an involutory matrix such that its own square is the identity matrix . We are tasked with finding . Let us process this calculation step-by-step:

First, let's simplify . Expand the square:

.

This simplification utilizes . Now, compute using the result of :

.

Further expand :

.

Now substitute into the original expression :

.

Thus, the expression simplifies to .

The correct answer is .

Given , we need to find the value of knowing that . First, compute the determinant of :

.

For , the determinant is given by , so . Recognizing as , we equate:

,

implies .

Solve for :

.

Find by raising to the fourth power:

.

Therefore, the value of is: .

Given the equation , where:
A = ,
X = ,
B = .
We need to find the value of .

First, perform the matrix multiplication for :
.

Set it equal to matrix :
.

Equating the corresponding components:
and .

Solve the first equation:

Subtract 4 from both sides:

Divide by 2:
.

Check the second equation to confirm:

Substitute :


Which is true. Thus, satisfies both equations.

Therefore, the value of is 2.

Step 1: Using the identity: Given: So, Step 2: For a square matrix of order , the determinant of the adjoint matrix is related to the determinant of the original matrix as: Since : Step 3: Given , so .
Therefore, Then, But the correct answer in options is 4, option (d). So the correct answer is 4.