**Question:**

**If find a matrix C such that 3A + 5B + 2C is a null matrix.**

**$A=\left[\begin{array}{cc}1 & 5 \\ 7 & 12\end{array}\right]$ and $B=\left[\begin{array}{ll}9 & 1 \\ 7 & 8\end{array}\right]$**

**Solution:**

Let’s consider a matrix C, such that

$C=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$

$3 A+5 B+2 C=O$

$\left[\begin{array}{cc}3 & 15 \\ 21 & 36\end{array}\right]+\left[\begin{array}{cc}45 & 5 \\ 35 & 40\end{array}\right]+\left[\begin{array}{cc}2 a & 2 b \\ 2 c & 2 d\end{array}\right]=\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]$

$\left[\begin{array}{ll}48+2 a & 20+2 b \\ 56+2 c & 76+2 d\end{array}\right]=\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]$

**Comparing the terms,**

**2a + 48 = 0 ⇒ a = -24**

**20 + 2b = 0 ⇒ b = -10**

**56 + 2c = 0 ⇒ c = -28**

**And,**

**76 + 2d = 0 ⇒ d = -38**

**Therefore, the matrix C is**

**$C=\left[\begin{array}{ll}-24 & -10 \\ -28 & -38\end{array}\right]$**