Test: Matrices - Civil Engineering (CE) MCQ

For any matrix A, (A + A’) is always symmetric and (A - A') is always skew-symmetric.

where P = 1/2(A+ A'), a symmetric matrix and Q = 1/212(A - A'), is a skew-symmetric matrix.

Caluclation:

∴ Required symmetric matrix Q
= 1/2 (B-B')

 which is the required answer.

If A is any matrix, the (A^T)^T = A.

We have,

 

∴ (A^T)^T 

= A

Given:
A is a symmetric matrix,
⇒ AT = A or a_ij = a_ji

So, by property of symmetric matrices

Given:
Order of A is 4 × 3, the order of B is 4 × 5 and the order of C is 7 x 3
The transpose of the matrix obtained by interchanging the rows and columns of the original matrix.
So, order of AT is 3 × 4 and order of CT is 3 × 7
Now,
A^TB = {3 x 4} {4 × 5} = 3 x 5
⇒ Order of ATB is 3 x 5
Hence order of (A^TB)^T is 5 x 3
Now order of (A^TB)^T C^T = {5 x 3} {3 x 7} = 5 x 7
∴ Order of (A^TB)^T C^T is 5 x 7

⇒ [(2x - 9)x + 8 x 4x] = 0

⇒ [2x² - 9x + 32x] = 0

⇒ 2x² + 23x = 0

⇒ x(2x + 23) = 0

⇒ x = 0 or - 23/2

Let D = -12 for the given matrix

(Taking 2 common from each row)

∴Det(A) =​ (2)³ × D

8x − 12 = −96​

Given that A is involuntary matrix,
⇒ A² = I
Now,
(I − A) (I + A) = I² – IA + AI − A² 
⇒ I – A + A – I         (∵ A² = I)
⇒ 0
∴ (I − A) (I + A) is zero matrix.

Sum of the eigen values of matrix is
= sum of diagonal values present in the matrix

Given: A² - 2A - I = 0
⇒ A.A - 2A = I
Post multiply by A^-1, we get
⇒ AAA^-1 - 2AA^-1 = IA^-1
⇒ AI - 2I = A^-1             [∵ AA ^-1 = A ^{- 1}A = I]
∴ A^-1 = A - 2
the inverse of A is A - 2

Now,

Hence Option 1st is correct answer.

Given

When we reflect a point in the x-y plane about the line y = x, we swap the x and y coordinates of the point.

If the original point is (x, y), after reflecting it across the line y = x, the new coordinates will be (y, x). In other words, the x-coordinate becomes the y-coordinate, and the y-coordinate becomes the x-coordinate.

To represent this transformation mathematically as a matrix, we multiply the coordinates of the point by the following matrix:

When we multiply this matrix by the point's coordinates (x, y), the result will be the new coordinates (y, x). So, the point (x, y) is transformed to (y, x) after the reflection.

Thus, this matrix swaps the coordinates of the point, which mirrors it over the line y = x.

Given: A = 

It is an upper triangular matrix. It's diagonal elements are eigen values.
The eigen values of the matrix are 1, 1, 1.
∴ Number of distinct eigen values = 1
Hence, option (B) is correct.

To understand the statements about traces of matrices, consider the properties of matrix traces:

• Trace is the sum of the diagonal elements of a square matrix.
• For matrices A (m × n) and B (n × m):
  • The products AB and BA are square matrices.
  • It is a known property that tr(AB) = tr(BA).
• For matrices C (n × n) and D (n × n):
  • Both CD and DC are also square matrices of the same size.
  • Similarly, tr(CD) = tr(DC).

Therefore, both statements about the trace properties are correct.