Questions

1. Question 1

   If $A=[a_{ij}]_{3\times 3}$, then show that:

   (i)

   $I_3A=A$

   (ii)

   $AI_3=A$

   SolutionTheory
2. Question 2

   If $A=\begin{bmatrix}0&-1&2\\3&2&1\\-1&0&4\end{bmatrix}$, $B=\begin{bmatrix}2&1&-1\\1&2&4\\-1&2&1\end{bmatrix}$ and $C=\begin{bmatrix}1&0&-2\\-1&5&0\\3&4&-1\end{bmatrix}$, then find:

   (i)

   $A-B$

   (ii)

   $B-C$

   (iii)

   $(A-B)-C$

   (iv)

   $A-(B-C)$

   SolutionTheory
3. Question 3

   If $A=\begin{bmatrix}-i&2i\\1&-i\end{bmatrix}$, $B=\begin{bmatrix}-i&1\\2i&1\end{bmatrix}$ and $C=\begin{bmatrix}2i&2i\\-i&i\end{bmatrix}$, then show that:

   (i)

   $(AB)C=A(BC)$

   (ii)

   $A(B+C)=AB+AC$

   SolutionTheory
4. Question 4

   If $A$ and $B$ are square matrices of the same order, then explain why in general:

   (i)

   $(A+B)^2\ne A^2+2AB+B^2$

   (ii)

   $(A-B)^2\ne A^2-2AB+B^2$

   (iii)

   $(A+B)(A-B)\ne A^2-B^2$

   SolutionTheory
5. Question 5

   If $A=\begin{bmatrix}-1&2&3\\1&0&2\\-3&5&3\end{bmatrix}$, then find $A+A^T$, $A-A^T$, $AA^T$, $A^TA$ and $(A^T)^T$.

   SolutionTheory
6. Question 6

   Solve the matrix equation $A^2-5A+4I-X=0$ if $A=\begin{bmatrix}2&0&1\\2&1&3\\1&-1&0\end{bmatrix}$.

   SolutionTheory
7. Question 7

   If $A$ and $B$ are two matrices such that $AB=B$ and $BA=A$, show that $A^2+B^2=A+B$.

   SolutionTheory