Login/Sign Up
Home » If , show that .
If A=\left[\begin{array}{cc}a b & b^{2} \\ -a^{2} & -a b\end{array}\right], show that A^{2}=0.
CBSE | Class 12 | Exercise 5C | Maths | Matrices | RS Aggarwal
If A=\left[\begin{array}{cc}a b & b^{2} \\ -a^{2} & -a b\end{array}\right], show that A^{2}=0.

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 show: \boldsymbol{A}^{2}=\boldsymbol{0}.
Now if number of columns in left matrix is equals to the number of rows in right matrixm, multiplication of two matrices is possible.

Next discuss the order of the matrices which are given.
The order of matrix \boldsymbol{A} is \mathbf{2} \times \mathbf{2}.
Thus the multiplications we can proceed as the multiplication A^{2} is possible.
\begin{aligned} A^{2}=A A &=\left(\begin{array}{cc} a b & b^{2} \\ -a^{2} & -a b \end{array}\right)\left(\begin{array}{cc} a b & b^{2} \\ -a^{2} & -a b \end{array}\right) \\ &=\left(\begin{array}{cc} a^{2} b^{2}-a^{2} b^{2} & a b^{3}-a b^{3} \\ -a^{3} b+a^{3} b & -a^{2} b^{2}+a^{2} b^{2} \end{array}\right)=\left(\begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array}\right) \end{aligned}

i

More Material