Test: Matrix - Mathematics MCQ

We know AA^-1 = I₂

first do AA^t then you will find
AA^t = I₄
[AA^T]^-1 = [I₄]^-1 = I₄

ANSWER :- c

Solution :- Let  A  be real skew symmetric and suppose  λ∈C  is an eigenvalue, with (complex) eigenvector  v . Then, denoting by  H  hermitian transposition,

λvHv=vH(λv)=vH(Av)=vH(−AHv)=−(vHAH)v=−(Av)Hv=−(λv)Hv=−λ¯vHv 

Since  vHv≠0 , as  v≠0 , we get

λ=−λ¯ 

so  λ  is purely imaginary (which includes 0). Note that the proof works the same for a antihermitian (complex) matrix.

With a completely similar technique you can prove that the eigenvalues of a Hermitian matrix (which includes real symmetric) are real.

Then also every minor or order 2 is also zero.
∴ rank (C) = 1

Highest possible rank of A= 2 ,as   Ax = b is an inconsistent system.

(P) Singular matrix → Determinant is zero
(Q) Non-square matrix → Determinant is not defined 
(R) Real symmetric → Eigen values are always real  
(S) Ortho gonal → Det erminant is always one 

Let A be a square matrix which is neither symmetric nor skew-symmetric, and A^T is its transpose. The sum and difference of these matrices are defined as:

• S = A + A^T
• D = A - A^T

1. Sum of A and A^T:

We are given:

S = A + A^T

To check if S is symmetric, we take the transpose of S:

S^T = (A + A^T)^T = A^T + A = A + A^T = S

Since S^T = S, the matrix S is symmetric.

2. Difference of A and A^T:

We are given:

D = A − A^T

To check if D is skew-symmetric, we take the transpose of D:

D^T = (A − A^T)^T = A^T − A = −(A − A^T) = −D

Since D = A - A^T, the matrix D is skew-symmetric.

Conclusion:

• S = A + A^T is symmetric.
• D = A - A^T is skew-symmetric.

Therefore, the correct answer is Option D: S is symmetric and D is skew-symmetric.

∴ Rank (A) = 2

Determining the Order of the Matrix Expression

To find the order of the given matrix expression P [(Xᵀ Y)⁻¹ Pᵀ]ᵀ, let's analyze each step systematically.

Step 1: Given Matrix Dimensions

X is a 4 × 3 matrix.
Y is a 4 × 3 matrix.
P is a 2 × 3 matrix.

Step 2: Compute Intermediate Matrices

1. Transpose of X (Xᵀ)
   Since X is 4 × 3, its transpose Xᵀ will be 3 × 4.

2. Product Xᵀ Y
   Xᵀ is 3 × 4.
   Y is 4 × 3.
   The product Xᵀ Y will have dimensions 3 × 3.

3. Inverse of Xᵀ Y [(Xᵀ Y)⁻¹]
   Since Xᵀ Y is a 3 × 3 square matrix, its inverse (if it exists) will also have dimensions 3 × 3.

4. Product P (Xᵀ Y)⁻¹
   P is 2 × 3.
   (Xᵀ Y)⁻¹ is 3 × 3.
   Their product P (Xᵀ Y)⁻¹ will have dimensions 2 × 3.

5. Transpose of P (Pᵀ)
   P is 2 × 3, so its transpose Pᵀ will have dimensions 3 × 2.

6. Product P (Xᵀ Y)⁻¹ Pᵀ
   P (Xᵀ Y)⁻¹ is 2 × 3.
   Pᵀ is 3 × 2.
   Their product P (Xᵀ Y)⁻¹ Pᵀ will have dimensions 2 × 2.

Step 3: Compute the Transpose of the Final Expression

The given expression is [P (Xᵀ Y)⁻¹ Pᵀ]ᵀ.
Since the matrix P (Xᵀ Y)⁻¹ Pᵀ is already 2 × 2, its transpose will also be 2 × 2.

Final Answer

The order of the matrix expression [P (Xᵀ Y)⁻¹ Pᵀ]ᵀ is 2 × 2.

A square matrix B is said to be skew-symmetric if its transpose is equal to the negative of the matrix itself. In other words:
B^T = – B
This means that the matrix B is skew-symmetric if when you transpose the matrix, you get the same matrix with all elements multiplied by -1.

Therefore, the correct answer is Option A: B^T = −B.