Types of Matrices MCQ Quiz - Objective Question with Answer for Types of Matrices - Download Free PDF

Concept:

1. Symmetric Matrix: Square matrix A is said to be symmetric if the transpose of matrix A is equal to matrix A itself ⇔ A^T = A or A’ = A

2. Skew-Symmetric Matrix or Anti-symmetric: Square matrix A is said to be skew-symmetric if transpose of matrix A is equal to negative of matrix A ⇔ A^T = −A

3. Hermitian Matrix:  A Hermitian matrix is a complex square matrix that is equal to its own conjugate transpose ⇔ \({{\bf{A}}^{\bf{T}}} = \;{\bf{\bar A}}\)

4. Skew-Hermitian: \({{\bf{A}}^{\bf{T}}} = - {\bf{\bar A}}\)

5. Let z = x + iy be a complex number.

• Conjugate of z =  = x – iy

Calculation:

Let A = \(\left[ {\begin{array}{*{20}{c}} 0&{ - 4 + i}\\ {4 + i}&0 \end{array}} \right]\)

Now, transpose of matrix A

 A^T = \(\left[ {\begin{array}{*{20}{c}} 0&{4 + i}\\ { - 4 + i}&0 \end{array}} \right]\)

⇒ A^T ≠ A

∴ given matrix is not symmetric

A^T = \(\left[ {\begin{array}{*{20}{c}} 0&{4 + i}\\ { - 4 + i}&0 \end{array}} \right] \ne - {\rm{A}}\)

∴ given matrix is not skew-symmetric.

Now, Conjugate of matrix A

\({\rm{\bar A}} = \;\left[ {\begin{array}{*{20}{c}} 0&{\overline { - 4 + i} }\\ {\overline {4 + i} }&0 \end{array}} \right] = \;\left[ {\begin{array}{*{20}{c}} 0&{ - 4 - i}\\ {4 - i}&0 \end{array}} \right] = \; - \left[ {\begin{array}{*{20}{c}} 0&{4 + i}\\ { - \;4 + i}&0 \end{array}} \right]\)
 

\( \Rightarrow {\rm{\bar A}} = - \;{{\rm{A}}^{\rm{T}}}\)

∴ given matrix is skew-hermitian matrix.

Explanation:

\(\rm A=\frac{1}{3}\begin{bmatrix}1&2&2\\\ 2&1&-2\\\ x&2&y\end{bmatrix}\) is orthogonal matrix.

So, AA^t = I

⇒ \(\frac{1}{3}\begin{bmatrix}1&2&2\\\ 2&1&-2\\\ x&2&y\end{bmatrix}\)\(\frac{1}{3}\begin{bmatrix}1&2&x\\\ 2&1&2\\\ 2&-2&y\end{bmatrix}\) = \(\begin{bmatrix}1&0&0\\\ 0&1&0\\\ 0&0&1\end{bmatrix}\)

⇒ \(\frac{1}{9}\begin{bmatrix}9&0&x+4+2y\\\ 0&9&2x+2-2y\\\ x+4+2y&2x+2-2y&x^2+4y+y^2\end{bmatrix}\) = \(\begin{bmatrix}1&0&0\\\ 0&1&0\\\ 0&0&1\end{bmatrix}\)

Comparing both sides we get

x + 4 + 2y = 0....(i)

2x + 2 - 2y = 0

⇒ x - y + 1 = 0...(ii)

Subtracting them 

3y = - 3 ⇒ y = -1

putting in (i) we get

x + 1 + 1 = 0

x = -2

Hence x + y = - 1 - 2 = -3

Option (4) is true.

Explanation:

A = (aij)2×2 and \(\rm a_{ij}=\left\{\begin{matrix}i^2+j^2;i\ne j\\\ i-j; i=j\end{matrix}\right\}\)

Then

Explanation:

Given AB = O

Let  A = \(\left[\begin{array}{lll} a & b \\ c & d \end{array}\right]\)

B = \(\left[\begin{array}{lll} p & q \\ r & s \end{array}\right]\)

Now AB = \(\left[\begin{array}{lll} a & b \\ c & d \end{array}\right] \left[\begin{array}{lll} p & q \\ r & s \end{array}\right] = \left[\begin{array}{lll} 0 & 0 \\ 0 & 0 \end{array}\right]\)

\(\left[\begin{array}{lll} ap+br & aq+bs \\ cp+dr & cq+ds \end{array}\right]= \left[\begin{array}{lll} 0 & 0 \\ 0 & 0 \end{array}\right]\)

Comparing both sides, we get

⇒ ap + br = 0...(1)

⇒ aq + bs = 0...(2)

⇒ cp + dr = 0...(3)

⇒ cq + ds = 0...(4)

From Eqs. (i) and (iii), we get:

⇒ (ad – bc)p = 0 and (ad – bc)r = 0 

From Eqs. (ii) and (iv), we get:

⇒ (ad – bc) q = 0 and (ad – bc)s = 0 

Now, if A is non-singular

ad – bc ≠ 0

So p = q = r = s = 0.

Then,  B is the null matrix.

∴ Option (c) is correct.

Concept -

(i) A diagonal matrix is a square matrix where all the elements outside the main diagonal are zero. The main diagonal is the set of elements where the row index equals the column index.

In other words, a diagonal matrix is a matrix in which all the off-diagonal elements are zero.

(ii) Identity matrix is \(\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1& 0 \\ 0 & 0 & 1 \\ \end{bmatrix} \)

Explanation -

We have a 3 x 3 diagonal matrix:

\(\begin{bmatrix} 2 & 0 & 0 \\ 0 & 1& 0 \\ 0 & 0 & 1 \\ \end{bmatrix} \)

In this matrix, all the elements off the main diagonal (i.e., non-diagonal elements) are zero. and diagonal elements are no zero.

Hence Option(4) is true.

Concept:

Properties of identity matrix:

If A is the square matrix of order n × n

• AI = IA = A
• Iⁿ = I           (Where n ∈ N)

Calculation:

Given

A² = I

Now, A3 + (A + I)2 - 9A - I² - A²

= A². A + A² + I² + 2AI - 9A - I² - A²

= I. A + I + I + 2AI - 9A - I - I           [∵ A2 = I and AI = IA = A]

= AI + 2AI - 9A 

= 3AI - 9A

= 3A - 9A

= - 6A

Concept:

• Elementary matrix: An elementary matrix is a matrix which differs from the identity matrix by one single elementary row operation
• An elementary matrix has each diagonal element 1.

Calculation:

Let's check option b,

Let E = \(\left[ {\begin{array}{*{20}{c}} 1&5&0\\ 0&1&0\\ 0&0&1 \end{array}} \right]\)

Apply R₁ → R₁ – 5R₂

\( \Rightarrow {\rm{E\;}} = \left[ {\begin{array}{*{20}{c}} 1&0&0\\ 0&1&0\\ 0&0&1 \end{array}} \right] = \;{{\rm{I}}_{3{\rm{\;}} \times 3}}\)

We can see that option B converted in to an identity matrix by one elementary operation.

Hence Option B is correct.

Shortcut Method:

We know that an elementary matrix has each diagonal element 1.

Only B has diagonal element is 1. (By Definition)

So, Option B is correct.

Concept:

A square matrix A = [a_ij]_{n × n} is said to be symmetric if A^T = A

A^T (Transpose) is obtained by changing rows to columns and columns to rows

Calculation:

Let A = \(\rm \begin{bmatrix} 3 & 0 & 0\\ 0 & 4 & 0\\ 0 & 0 & 0 \end{bmatrix}\)

 A^T = \(\rm \begin{bmatrix} 3 & 0 & 0\\ 0 & 4 & 0\\ 0 & 0 & 0 \end{bmatrix}\) = A

 A is  a symmetric matrix

Concept:

Identity matrix: A square matrix in which elements in the main diagonal are all '1' and the rest are all zero is called an identity matrix or unit matrix.

Thus, the square matrix A = [\(a_{ij}\)], if \(a_{ij}=\left\{\begin{matrix}1,if \hspace{3mm} i= j \\ 0,if \hspace{3mm} i\neq j \end{matrix}\right.\)

e.g., \(\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\), \(\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix}\)are identity matrices of order 2 and 3 respectively.

For any natural number n, Iⁿ = I.ij]nneeeIⁿ_{\(I^n = I\)}

Calculation:

We have, A² =I

Also A and i are commutative, so we can expand (A + I)ⁿ using the expansion of (a+ b)ⁿ.

∴ (A – I)3 + (A + I)3 – 7A

= A³ - 3A² + 3A - I³ + A³ + 3A² + 3A + I² - 7A

= 2A³ + 6A - 7A

= 2A² ⋅A + 6A - 7A

= 2I ⋅A + 6A - 7A

= 2A + 6A - 7A

= 8A - 7A

= A

CONCEPT:

Symmetric Matrix:

Any real square matrix A = (a_ij) is said to be symmetric matrix if and only if a_ij = a_ji, ∀ i and j or in other words we can say that if A is a real square matrix such that A = A^t then A is said to be a symmetric matrix.

Skew-symmetric Matrix:

Any real square matrix A = (a_ij) is said to be skew-symmetric matrix if and only if a_ij = - a_ji, ∀ i and j or in other words we can say that if A is a real square matrix such that A = - A^t then A is said to be a skew-symmetric matrix.

Properties of Transpose of a Matrix:

• If A is a matrix of order m × n, then (A^t)^t = A
• If k ∈ R is a scalar and A is a matrix of order m × n, then (k × A)^t = k × A^t
• If A and B are matrices of same order m × n, then (A ± B)^t = A^t ± B^t.

CALCULATION:

Given: A is skew symmetric matrix

As we know that, if A is a skew symmetric matrix i.e A = - At 

⇒ (A²)^t = (A^t)²

∵ A is skew symmetric matrix

⇒ (A2)t = (- A)² = A²

So, A² is a symmetric matrix.

Hence, option D is the correct answer.

Concept:

1. Symmetric Matrix: Square matrix A is said to be symmetric if the transpose of matrix A is equal to matrix A itself ⇔ A^T = A or A’ = A

2. Skew-Symmetric Matrix or Anti-symmetric: Square matrix A is said to be skew-symmetric if transpose of matrix A is equal to negative of matrix A ⇔ A^T = −A

3. Hermitian Matrix:  A Hermitian matrix is a complex square matrix that is equal to its own conjugate transpose ⇔ \({{\bf{A}}^{\bf{T}}} = \;{\bf{\bar A}}\)

4. Skew-Hermitian: \({{\bf{A}}^{\bf{T}}} = - {\bf{\bar A}}\)

5. Let z = x + iy be a complex number.

• Conjugate of z =  = x – iy

Calculation:

Let A = \(\left[ {\begin{array}{*{20}{c}} 0&{ - 4 + i}\\ {4 + i}&0 \end{array}} \right]\)

Now, transpose of matrix A

 A^T = \(\left[ {\begin{array}{*{20}{c}} 0&{4 + i}\\ { - 4 + i}&0 \end{array}} \right]\)

⇒ A^T ≠ A

∴ given matrix is not symmetric

A^T = \(\left[ {\begin{array}{*{20}{c}} 0&{4 + i}\\ { - 4 + i}&0 \end{array}} \right] \ne - {\rm{A}}\)

∴ given matrix is not skew-symmetric.

Now, Conjugate of matrix A

\({\rm{\bar A}} = \;\left[ {\begin{array}{*{20}{c}} 0&{\overline { - 4 + i} }\\ {\overline {4 + i} }&0 \end{array}} \right] = \;\left[ {\begin{array}{*{20}{c}} 0&{ - 4 - i}\\ {4 - i}&0 \end{array}} \right] = \; - \left[ {\begin{array}{*{20}{c}} 0&{4 + i}\\ { - \;4 + i}&0 \end{array}} \right]\)
 

\( \Rightarrow {\rm{\bar A}} = - \;{{\rm{A}}^{\rm{T}}}\)

∴ given matrix is skew-hermitian matrix.

Concept:

Consider a matrix A is skew-symmetric, then A^T = −A
and A is symmetric, then A^T = A

Calculation:

Since, A is skew-symmetric.
A^T = −A
Since, A is symmetric.
A^T = A
⇒ −A = A
⇒2A = O
⇒A = O
Hence, A is a null matrix.

Hence, option (4) is correct.

Concept:

Singular Matrix: Any square matrix of order n is said to be singular if |A| = 0.

Involuntary Matrix: Any square matrix of order n is said to be an involuntary matrix if A² = I, where I is the identity matrix of order n.

Nilpotent Matrix: Any square matrix of order n is said to be nilpotent matrix if there exist least positive integer m such that A^m = O, where O is the null matrix of order n.

Idempotent Matrix: Any square matrix of order n is said to be an idempotent matrix if A² = A.

Calculation:

Given: \(A = \left[ {\begin{array}{*{20}{c}} 0&1\\ 1&0 \end{array}} \right],\)

\(\Rightarrow \;{A^2} = \;\left[ {\begin{array}{*{20}{c}} 0&1\\ 1&0 \end{array}} \right] \times \;\left[ {\begin{array}{*{20}{c}} 0&1\\ 1&0 \end{array}} \right] \\= \;\left[ {\begin{array}{*{20}{c}} 1&0\\ 0&1 \end{array}} \right] = I\)

Concept:

• For symmetric matrices, A = A' and B = B'
• For skew-symmetric matrices, A = - A'
• (A ± B)' = A' ± B'
• (AB)' = B'A'

Calculation:

Given: A and B are symmetric matrices

As we know that for symmetric matrices, we have A = A' and B = B'

(AB - BA)' = (AB)' - (BA)' --------(∵ (A ± B)' = A' ± B')

⇒ (AB - BA)' = B'A' - A'B' --------(∵ (AB)' = B'A')

⇒ (AB - BA)' = BA - AB ----------(∵ A = A' and B = B')

⇒ (AB - BA)' = - (AB - BA)

Hence Option 3 is correct.

Explanation:

Here we have to fill 4 entries it means we can form 2 × 2 matrices, 4 × 1 and 1 × 4 matrices

In 2 × 2 matrices, there are four places 

And each place has two ways to fill (1, 2) 

So, The total number of ways to fill the entries =  2 × 2 × 2 × 2 = 16.

In 4 × 1 matrices, there are four places

And each place has two ways to fill (1, 2) 

So, The total number of ways to fill the entries =  2 × 2 × 2 × 2 = 16.

Similarly in case of 1 × 4 matrices total number of ways = 16.

∴ Total number of matrices will be 48.