Matrices

It is given that A=

To show: (aI+bA)ⁿ=aⁿI+na^n-1bA
We shall prove the result by using the principle of mathematical induction.

For n = 1, we have:
P(1):(aI+bA)=aI+ba⁰A=aI+bA

Therefore, the result is true for n =1.
Let the result be true for n = k.
That is,
P(k):(aI+bA)^k=akI+ka^k-1bA
Now, we prove that the result is true for n = k + 1.
Consider:
(aI+bA)^k+1=(aI+bA)^k(aI+bA)
=(a^kI+ka^k-1bA)(aI+bA)
=a^k+1+ka^kbAI+a^kbIA+ka^k-1b²A²
=a^k+1I+(k+1)a^kbA+ka^k-1b²A² ...(1)

Now A²= = = O

From (1), we have:
(aI+bA)^k+1=a^k+1I+(k+1)a^kbA+O
=a^k+1I+(k+1)a^kbA

Therefore, the result is true for n = k + 1.

Thus, by the principle of mathematical induction, we have:

(aI+bA)ⁿ=aⁿI+na^n-1bA where A= ,n∈N

Step 1: {Matrix inversion property}
If is given, the original matrix is the reciprocal of the scalar multiple.
Step 2: {Calculate }
Given , we compute by multiplying the inverse by , so:

Step 3: {Verify the options}
The correct matrix is option (B).

Step 1: Understanding the property of .
The function satisfies , which means is an odd function.
Step 2: Integral over an asymmetric interval.
The integral can be related to the integral over the negative interval using the property of odd functions:
Step 3: Analyzing the given options.
Option (D) correctly expresses the relationship between the integral over and for odd functions.
Step 4: Conclusion.
The integral equals . Thus, the correct answer is (D). {10pt}

Step 1: Understanding the expression. The given expression represents the sum of products of elements of the second row with the cofactors of the corresponding elements from the first row.

Step 2: Determinant Property. By the cofactor expansion property: since this is equivalent to the determinant expansion along a different row of the same matrix.

Conclusion: Thus, the required value is , which corresponds to option .

Step 1: Characteristic Equation. We substitute into the given equation: Calculating :

Using matrix identity , substituting in the equation and solving for , we obtain:

Conclusion: Thus, the required value is , which corresponds to option .

Step 1: Understand the series : The given series is , which is an infinite series. It can be expressed as: if the matrix is invertible.

Step 2: Compute : The identity matrix is:

So, .

Step 3: Check if is invertible: The determinant of is:
Since the determinant is non-zero, is invertible.

Step 4: Verify the series sum: The inverse of is:

Thus, the sum of the series is:

Conclusion: The correct option is

Step 1: Recall the property of skew-symmetric matrices
For a skew-symmetric matrix , .
Step 2: Analyze
Taking the transpose: Thus, is symmetric, as its transpose equals itself.

Step 1: Compute the matrix product. Given matrices and , compute : = = 7I
Since , it follows that .

Step 2: Compute .

Step 3: Convert system into matrix form. The given system can be written as: where , ,

Thus,

Step 4: Compute .

Multiplying the matrices:

Conclusion: Thus, the solution to the given system is:

To solve the problem, we are given a matrix and we need to compute its inverse . Then, using the inverse, solve the system of linear equations.

1. Writing the Matrix Form of the System:
We rewrite the system of equations as a matrix equation , where:

2. Finding the Inverse :
We calculate using the adjoint method or row reduction. The inverse is given as:


Computing the determinant of :


So the determinant is 1.
Now computing the adjugate (cofactor matrix transpose), we get:

3. Solving for :
Now multiply with :


Performing the multiplication:

Final Answer:
The solution to the system is:

A scalar matrix is a matrix in which all the diagonal elements are equal, and all the off-diagonal elements are zero.
The given matrix is: This matrix is not a scalar matrix because its diagonal elements are not all equal. Thus, we rule out the scalar matrix option.
Next, we observe that a symmetric matrix is one where , and a skew-symmetric matrix is one where .
The given matrix is neither symmetric nor skew-symmetric, as it is not equal to its transpose and also not equal to the negative of its transpose. Therefore, the correct answer is that is a scalar matrix.

• Matrix multiplication means is multiplied by the transpose of . For this multiplication to be defined, the number of columns in must equal the number of rows in . Since (the transpose of ) has the number of rows equal to the number of columns in , must have columns.
• The matrix resulting from the transpose of will have dimensions if is .
• For multiplication, the number of columns in must match the number of rows in . If is , then must be for the multiplication to be defined. However, since is , it suffices to consider only the requirement that has dimensions to satisfy both and being defined.

To solve the problem, we need to identify which type of matrix can be both symmetric and skew-symmetric.

1. Understanding Symmetric and Skew-Symmetric Matrices:
A matrix is symmetric if:



A matrix is skew-symmetric if:


Now, if a matrix is both symmetric and skew-symmetric, then:


This implies:


So, the only matrix that satisfies both conditions is the null matrix (all elements are zero).

2. Evaluating Each Option:
(A) Unit Matrix → Not possible. It's symmetric but not skew-symmetric.
(B) Diagonal Matrix → Could be symmetric, but not necessarily skew-symmetric.
(C) Null Matrix → Satisfies both and . → Correct
(D) Row Matrix → Not necessarily square, and thus not even eligible for symmetric/skew-symmetric.

3. Conclusion:
The null matrix is the only matrix that is both symmetric and skew-symmetric.

Final Answer:
The correct answer is Null Matrix.

To solve the problem, we need to analyze the nature of the matrix expression when and are square matrices of the same order.

1. Take the Transpose of the Expression:
Let’s compute the transpose of :


Using the identity , we get:

, and
So:

2. Interpretation:
Since the transpose of the expression equals the negative of the expression, it satisfies the condition for a skew-symmetric matrix:

is skew-symmetric

3. Conclusion:
The matrix is skew-symmetric.

Final Answer:
The correct option is (B) skew-symmetric matrix.

• Diagonal Matrix: A matrix in which the entries outside the main diagonal are all zero. For matrix , it looks like this:

• This is indeed a diagonal matrix.
• Scalar Matrix: A special type of diagonal matrix where all diagonal elements are equal.

• The Reason (R) states that if a diagonal matrix has all non-zero elements equal, it becomes a scalar matrix. This is true based on the definition of a scalar matrix.

To determine whether the matrix products and are defined, we need to check the dimensions of the matrices.
- Matrix has dimensions (2 rows and 3 columns).
- Matrix has dimensions (3 rows and 2 columns).
For the product to be defined, the number of columns of must match the number of rows of . In this case, has 3 columns and has 3 rows, so is defined.
The resulting matrix will have dimensions . For the product , the number of columns of must match the number of rows of .
However, has 2 columns and has 2 rows, so is not defined. Hence, only is defined.

A scalar matrix is a diagonal matrix in which all the diagonal elements are equal. In this case, the matrix has the diagonal elements , , and , which are not equal. Therefore, this is not a scalar matrix. However, it is a diagonal matrix and the definition of a scalar matrix requires all diagonal elements to be the same.
So, this is not a scalar matrix, making it a symmetric matrix as it satisfies for this case. The correct answer is (A).