#### Please solve RD Sharma class 12 chapter 4 Algebra of Matrices exercise Fill in the blank question 41 maths textbook solution

$A=\left[\begin{array}{ll} \frac{1}{3} & \frac{1}{3} \\ \\ \frac{2}{3} & \frac{1}{3} \end{array}\right]$

Hint:

For that we must aware with basic operations of matrix

Given:

\begin{aligned} &A+B=\left[\begin{array}{ll} 1 & 0 \\ 1 & 1 \end{array}\right] \\ &A-2 B=\left[\begin{array}{cc} -1 & 1 \\ 0 & -1 \end{array}\right] \end{aligned}

Solution:

\begin{aligned} &A+B=\left[\begin{array}{ll} 1 & 0 \\ 1 & 1 \end{array}\right]\; \; \; \; \; \; .....(i) \\ &A-2 B=\left[\begin{array}{cc} -1 & 1 \\ 0 & -1 \end{array}\right] \; \; \; \; \; \; .....(ii) \end{aligned}

\begin{aligned} &A+B+A-2 B=\left[\begin{array}{ll} 1 & 0 \\ 1 & 1 \end{array}\right]+\left[\begin{array}{cc} -1 & 1 \\ 0 & -1 \end{array}\right]\\ &2 A-B=\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right] \; \; \; \; \; \; .....(ii) \end{aligned}

Lets add both (iii) and (i) equation

\begin{aligned} &3 A=\left[\begin{array}{ll} 1 & 1 \\ 2 & 1 \end{array}\right] \\ &A=\frac{1}{3}\left[\begin{array}{ll} 1 & 1 \\ 2 & 1 \end{array}\right] \\ &A=\left[\begin{array}{ll} \frac{1}{3} & \frac{1}{3} \\ \\ \frac{2}{3} & \frac{1}{3} \end{array}\right] \end{aligned}