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CodeVID-M12-03-TRN-01
Transpose, Symmetric & Invertible Matrices — Assignment
Chapter: Matrices
Topic: Transpose, Symmetric & Invertible Matrices
Maximum Marks: 35
Time: 75 minutes
Name: ____________________ Roll No.: __________ Date: ____________

General Instructions

  • All questions are compulsory.
  • Section A carries 1 mark each, Section B 2 marks, Section C 3 marks and Section D 5 marks.
  • Show all working for Sections B, C and D. Only final answers are given at the end — for full solutions, raise your doubts with your teacher.
Section A — Multiple Choice Questions 5 × 1 = 5 marks
1.
$(AB)^{T}=$
  • A.$A^{T}B^{T}$
  • B.$B^{T}A^{T}$
  • C.$AB$
  • D.$BA$
2.
A skew-symmetric matrix has diagonal entries:
  • A.all $1$
  • B.all $0$
  • C.equal non-zero
  • D.unrestricted
3.
The symmetric part of $A$ is:
  • A.$\tfrac12(A-A^{T})$
  • B.$\tfrac12(A+A^{T})$
  • C.$A^{T}$
  • D.$A-A^{T}$
4.
$A$ is invertible iff:
  • A.$A$ is symmetric
  • B.$\det A=0$
  • C.$\det A\ne0$
  • D.$A=A^{T}$
5.
$(A^{T})^{T}=$
  • A.$A^{-1}$
  • B.$A$
  • C.$A^{T}$
  • D.$I$
Section B — Short Answer (2 marks) 4 × 2 = 8 marks
6.
Find $A^{T}$ for $A=\begin{bmatrix}1&2&3\\4&5&6\end{bmatrix}$.
7.
Is $A=\begin{bmatrix}0&3\\-3&0\end{bmatrix}$ symmetric or skew-symmetric?
8.
Find $A^{-1}$ for $A=\begin{bmatrix}3&0\\0&5\end{bmatrix}$.
9.
If $A$ is symmetric, what is $A^{T}$?
Section C — Short Answer (3 marks) 4 × 3 = 12 marks
10.
Express $A=\begin{bmatrix}2&3\\1&4\end{bmatrix}$ as a sum of a symmetric and a skew-symmetric matrix.
11.
Verify $(AB)^{T}=B^{T}A^{T}$ for $A=\begin{bmatrix}1&2\\0&1\end{bmatrix},\ B=\begin{bmatrix}1&0\\3&1\end{bmatrix}$.
12.
Show that $A+A^{T}$ is symmetric for any square $A$.
13.
Find $A^{-1}$ for $A=\begin{bmatrix}2&3\\1&4\end{bmatrix}$.
Section D — Long Answer (5 marks) 2 × 5 = 10 marks
14.
Express $A=\begin{bmatrix}1&2&4\\6&8&1\\3&5&7\end{bmatrix}$ as the sum of a symmetric and a skew-symmetric matrix.
15.
If $A=\begin{bmatrix}2&3\\1&2\end{bmatrix}$, verify $A\,(\operatorname{adj}A)=|A|\,I$ and find $A^{-1}$.

Answer Key

Section A — Multiple Choice Questions
  1. (B) $B^{T}A^{T}$
  2. (B) all $0$
  3. (B) $\tfrac12(A+A^{T})$
  4. (C) $\det A\ne0$
  5. (B) $A$
Section B — Short Answer (2 marks)
  1. $\begin{bmatrix}1&4\\2&5\\3&6\end{bmatrix}$.
  2. Skew-symmetric.
  3. $\begin{bmatrix}\tfrac13&0\\0&\tfrac15\end{bmatrix}$.
  4. $A^{T}=A$.
Section C — Short Answer (3 marks)
  1. $\begin{bmatrix}2&2\\2&4\end{bmatrix}+\begin{bmatrix}0&1\\-1&0\end{bmatrix}$.
  2. Both equal $\begin{bmatrix}7&3\\2&1\end{bmatrix}$.
  3. $(A+A^{T})^{T}=A^{T}+A=A+A^{T}$ — symmetric.
  4. $\tfrac15\begin{bmatrix}4&-3\\-1&2\end{bmatrix}$.
Section D — Long Answer (5 marks)
  1. Symmetric part $\begin{bmatrix}1&4&\tfrac72\\4&8&3\\\tfrac72&3&7\end{bmatrix}$, skew part $\begin{bmatrix}0&-2&\tfrac12\\2&0&-2\\-\tfrac12&2&0\end{bmatrix}$.
  2. $|A|=1$; $A^{-1}=\begin{bmatrix}2&-3\\-1&2\end{bmatrix}$.
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