Matrices

### If the matrix A is both symmetric and skew-symmetric, show that A is a zero matrix.

Solution: The transpose of the matrix is an operation of making interchange of elements by the rule on positioned element $a_{j i}$ shifted to new position $a_{j i}$. The symmetric matrix is defined...

### If and , find .

Solution: We have $A=\left(\begin{array}{ll}2 & 3 \\ 1 & 2\end{array}\right)$. Now addition/subtraction of two matrices is possible if order of both the matrices are same and multiplication...

### If A and B are symmetric matrices of the same order, show that (AB – BA) is a skew symmetric matrix.

Solution: We have $A$ and $B$ are symmetric matrices. Therefore $A^{T}=A$ and $B^{T}=B$ The transpose of the matrix is an operation of making interchange of elements by the rule on positioned...

### If , show that .

Solution: We have $A=\left(\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha\end{array}\right)$. The transpose of the matrix is an operation of making interchange of...

### If and , find a matrix such that

Solution: We have $A=\left(\begin{array}{ll}4 & 2 \\ 1 & 3\end{array}\right), B=\left(\begin{array}{cc}-2 & 1 \\ 3 & 2\end{array}\right)$ and $3 A-2 B+X=0$ We can have the addition...

### If and , find a matrix such that .

Solution: We have $A=\left(\begin{array}{cc}2 & -3 \\ 4 & 5\end{array}\right), B=\left(\begin{array}{cc}-1 & 2 \\ 0 & 3\end{array}\right)$ and $A+2 B+X=0$. We can have the addition...

### If , and show that is skew-symmetric

Solution: We have $\left(\begin{array}{ll}2 & 3 \\ 4 & 5\end{array}\right)$. The transpose of the matrix is an operation of making interchange of elements by the rule on positioned element...

### If , show that is symmetric

Solution: We have $\left(\begin{array}{ll}4 & 5 \\ 1 & 8\end{array}\right)$. The transpose of the matrix is an operation of making interchange of elements by the rule on positioned element...

### Find the value of and for which

Solution: We have $\left(\begin{array}{cc}x & y \\ 3 y & x\end{array}\right)\left(\begin{array}{l}1 \\ 2\end{array}\right)=\left(\begin{array}{l}3 \\ 5\end{array}\right)$. Use the...

### Find the value of and for which

Solution: We have $\left(\begin{array}{cc}2 & -3 \\ 1 & 1\end{array}\right)\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{l}1 \\ 3\end{array}\right)$. Use the...

### If then find the least value of for which .

Solution: We have $A=\left(\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha\end{array}\right)$ Use the addition rule and get $A+A^{T}=I_{2}$ as follow:...

### If and , find the matrix such that is a zero matrix

Solution: We have $A=\left(\begin{array}{cc}1 & -5 \\ -3 & 2 \\ 4 & -2\end{array}\right) ; B=\left(\begin{array}{cc}3 & 1 \\ 2 & -1 \\ -2 & 3\end{array}\right) .$ and...

### If , find the values of .

Solution: We have $\left(\begin{array}{cc}x & 3 x-y \\ 2 x+z & 3 y-w\end{array}\right)=\left(\begin{array}{ll}3 & 2 \\ 4 & 7\end{array}\right)$. Now from the equality of matrices we...

### If , find the values of and .

Solution: We have $x\left(\begin{array}{l}2 \\ 3\end{array}\right)+y\left(\begin{array}{c}-1 \\ 1\end{array}\right)=\left(\begin{array}{c}10 \\ 5\end{array}\right)$. Use the addition rule and get...

### Express the matrix as sum af two matrices such that and is symmetric and the other is skew-symmetric.

Solution: Given that $A=\left[\begin{array}{lll}3 & 2 & 5 \\ 4 & 1 & 3 \\ 0 & 6 & 7\end{array}\right]$, to express as sum of symmetric matrix $P$ and skew symmetric matrix...

### Express the matrix A as the sum of a symmetric and a skew-symmetric matrix, where .

Solution: Given that $A=\left[\begin{array}{ccc}3 & -1 & 0 \\ 2 & 0 & 3 \\ 1 & -1 & 2\end{array}\right]$, to express as sum of symmetric matrix $P$ and skew symmetric matrix...

### Express the matrix as the sum of a symmetric and a skew-symmetric matrix.

Solution: Given that $\mathrm{A}=\left[\begin{array}{ccc}-1 & 5 & 1 \\ 2 & 3 & 4 \\ 7 & 0 & 9\end{array}\right]$, to express as sum of symmetric matrix $\mathrm{P}$ and skew...

### Express the matrix as the sum of a symmetric matrix and a skew-symmetric matrix.

Solution: Given that $\mathrm{A}=\left[\begin{array}{rr}3 & -4 \\ 1 & -1\end{array}\right]$,to express as the sum of symmetric matrix $\mathrm{P}$ and skew symmetric matrix Q. $A=P+Q$ Where...

### If , find

Solution: We have $A=\left(\begin{array}{ccc}2 & 1 & 2 \\ 1 & 0 & 2 \\ 0 & 2 & -4\end{array}\right), B=\left(\begin{array}{lll}x & 4 & 1\end{array}\right)$ and...

### If , find

Solution: We have $A=\left(\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6 \\ 3 & 2 & 5\end{array}\right), B=\left(\begin{array}{lll}1 & x & 1\end{array}\right)$ and...

### If , show that .

Solution: We have $\boldsymbol{A}=\left(\begin{array}{cc}\boldsymbol{a} \boldsymbol{b} & \boldsymbol{b}^{2} \\ -\boldsymbol{a}^{2} & -\boldsymbol{a} \boldsymbol{b}\end{array}\right) .$ To...

### If and . show that is a zero matrix.

Solution: We have $\boldsymbol{A}=\left(\begin{array}{ccc}\mathbf{0} & \boldsymbol{c} & -\boldsymbol{b} \\ -\boldsymbol{c} & \mathbf{0} & \boldsymbol{a} \\ \boldsymbol{b} &... read more Solution: We have$\boldsymbol{A}=\left(\begin{array}{ccc}\mathbf{2} & -\mathbf{3} & -\mathbf{5} \\ -\mathbf{1} & \mathbf{4} & \mathbf{5} \\ \mathbf{1} & -\mathbf{3} &...

### Compute AB and BA, which ever exists when

Solution: We have $\boldsymbol{A}=\left(\begin{array}{ccc}\mathbf{0} & \mathbf{1} & -\mathbf{5} \\ \mathbf{2} & \mathbf{4} & \mathbf{0}\end{array}\right)$ and...

### Compute AB and BA, which ever exists when

Solution: We have $\boldsymbol{A}=\left(\begin{array}{cc}-\mathbf{1} & \mathbf{1} \\ -\mathbf{2} & \mathbf{2} \\ -\mathbf{3} & \mathbf{3}\end{array}\right)$ and...

### Compute AB and BA, which ever exists when

Solution: We have $\boldsymbol{A}=\left(\begin{array}{cc}\mathbf{2} & \mathbf{- 1} \\ \mathbf{3} & \mathbf{0} \\ \mathbf{- 1} & \mathbf{4}\end{array}\right)$ and...

### If then write the value of

Solution: $\text { If }\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]=\left[\begin{array}{ll} e & f \\ g & h \end{array}\right]$ Therefore $a=e, b=f, c=g, d=h$ It is given...

### Construct a matrix whose elements are given by

Solution: It is a (3 $x 4)$ matrix. Therefore, it has 3 rows and 4 columns. Given that $a_{i j}=\frac{|-a \|+| l}{2}$ Therefore, $a_{11}=1 . a_{12}=\frac{1}{2}, a_{13}=0, a_{13}=\frac{1}{2}$...