Matrix

The given determinant is:

Using the formula for the determinant of a matrix:

We calculate:

Expanding each term:

Subtracting the two products:

Hence, the value of the determinant is , and the correct answer is (B).

Expand the left-hand side of the given equation:

Equating both sides:

Cancel and :

Rearranging:

Therefore, the correct answer is (B) .

Step 1: Definition of an identity matrix.

An identity matrix is a square matrix in which all the diagonal elements are , and all off-diagonal elements are . Mathematically:

Step 2: Analyze each option.

(A) if and if : This is incorrect because it contradicts the definition of an identity matrix.
(B) : This is incorrect because an identity matrix has for all off-diagonal elements.
(C) : This is incorrect because it implies all elements are , which is not an identity matrix.
(D) if and if : This is correct, as it matches the definition of an identity matrix.

Final Answer:

Step 1: Understand the given data
Matrix X is of order 2 × 3, meaning it has 2 rows and 3 columns.
Matrix Y is of order m × n (unknown, we have to find m and n).
The product matrix XY is of order 2 × 5.

Step 2: Condition for matrix multiplication
For two matrices X and Y to be multiplied (XY), the number of columns of X must be equal to the number of rows of Y.
Given that X has 3 columns, so m = 3.

Step 3: Determine the order of matrix Y
Since Y has order m × n, and m = 3, Y is of order 3 × n.

Step 4: Determine the number of columns n of Y
The product matrix XY has order 2 × 5, which means the number of columns of Y is 5.
So, n = 5.

Step 5: Calculate the number of elements in matrix Y
Number of elements in matrix Y = number of rows × number of columns = 3 × 5 = 15.

Final Answer: 15

Step 1: Use the property of matrix inverses.

The product of a matrix and its inverse is the identity matrix:

where is the identity matrix.

Step 2: Multiply and .

Compute the product :

Step 3: Analyze each element of the product.

From the first row of the product:

Simplify each equation:

1. implies implies
2. implies
3. implies implies a = 1.

From the third row of the product:

Simplify each equation:

1. implies b = 5.
2. implies y = -4.

Step 4: Compute .

Substitute the values , , , and :

Simplify:

Final Answer:

From the equality of the matrices:

Step 1: Use the relationship between and
The given equations are: These are the sum and product of the roots of a quadratic equation. Let the quadratic equation be:

Substitute and :

Factorize the equation: Thus, and (or vice versa).

Step 2: Compute
Substitute and : Simplify:

We can use the formula for the product of two binomials: This result follows from the distributive property.
Now, since matrix multiplication is not commutative, we cannot assume .
Therefore, the correct expression is:

Step 1: Understanding Symmetric Matrices

A matrix is symmetric if:

This means that elements across the main diagonal must be equal:

Step 2: Equating Corresponding Elements

The given matrix is:

Since it's symmetric, we equate the off-diagonal elements:

2. 
   Substituting :
   

Step 3: Compute