Find the value of a, b, c and d from the equation:`[[a-b],[2a+c][2a-b,3c+d]]=[[-1 ,5 ],[0 ,13]]`

Explanation

To solve the equation given by the matrix:

\[ \begin{pmatrix} a-b & 2a+c \\ 2a-b & 3c+d \end{pmatrix} = \begin{pmatrix} -1 & 5 \\ 0 & 13 \end{pmatrix} \]

we will equate the corresponding elements of the matrices. This gives us the following equations:

1. \( a - b = -1 \) (Equation 1)
2. \( 2a + c = 5 \) (Equation 2)
3. \( 2a - b = 0 \) (Equation 3)
4. \( 3c + d = 13 \) (Equation 4)

Now, we will solve these equations step by step.

### Step 1: Solve Equations 1 and 3

From Equation 1:
\[ a - b = -1 \implies a = b - 1 \quad \text{(Equation 5)} \]

From Equation 3:
\[ 2a - b = 0 \implies 2a = b \implies b = 2a \quad \text{(Equation 6)} \]

### Step 2: Substitute Equation 5 into Equation 6

Substituting Equation 5 into Equation 6:
\[ b = 2(b - 1) \]
Expanding this gives:
\[ b = 2b - 2 \]
Rearranging gives:
\[ 2 = 2b - b \implies b = 2 \]

### Step 3: Find the value of \( a \)

Substituting \( b = 2 \) back into Equation 5:
\[ a = 2 - 1 = 1 \]

### Step 4: Solve for \( c \) using Equation 2

Now, substituting \( a = 1 \) into Equation 2:
\[ 2(1) + c = 5 \implies 2 + c = 5 \implies c = 5 - 2 = 3 \]

### Step 5: Solve for \( d \) using Equation 4

Now, substituting \( c = 3 \) into Equation 4:
\[ 3(3) + d = 13 \implies 9 + d = 13 \implies d = 13 - 9 = 4 \]

### Final Values

Thus, the values of \( a, b, c, \) and \( d \) are:
\[ a = 1, \quad b = 2, \quad c = 3, \quad d = 4 \]