If $2\left[\begin{array}{ll}a & d \\ b & c\end{array}\right]+3\left[\begin{array}{cc}1 & -1 \\ 0 & 2\end{array}\right]=3\left[\begin{array}{ll}3 & 5 \\ 4 & 6\end{array}\right]$, then the value of $|a+b-c-d|$ is

6

The correct answer is Option (3) → 6

The given equation is:

$2\begin{bmatrix} a & d \\ b & c \end{bmatrix} + 3\begin{bmatrix} 1 & -1 \\ 0 & 2 \end{bmatrix} = 3\begin{bmatrix} 3 & 5 \\ 4 & 6 \end{bmatrix}$

Step 1: Perform Scalar Multiplication

Multiply each matrix by its respective scalar:

$\begin{bmatrix} 2a & 2d \\ 2b & 2c \end{bmatrix} + \begin{bmatrix} 3 & -3 \\ 0 & 6 \end{bmatrix} = \begin{bmatrix} 9 & 15 \\ 12 & 18 \end{bmatrix}$

Step 2: Set up Equations for Each Element

By adding the matrices on the left and equating them to the matrix on the right, we get four linear equations:

1. Top-left element: $2a + 3 = 9 ⇒ 2a = 6 ⇒\mathbf{a = 3}$
2. Top-right element: $2d - 3 = 15 ⇒2d = 18 ⇒\mathbf{d = 9}$
3. Bottom-left element: $2b + 0 = 12 ⇒2b = 12 ⇒\mathbf{b = 6}$
4. Bottom-right element: $2c + 6 = 18 ⇒2c = 12 ⇒\mathbf{c = 6}$

Step 3: Calculate $|a + b - c - d|$

Now, substitute the values into the required expression:

$|a + b - c - d| = |3 + 6 - 6 - 9|$

$= |9 - 15|$

$= |-6|$

$= 6$