If `[(2,-1), (1, 0),(-3, 4)]A=[(-1, -8, -10), (1, -2, -5), (9, 22, 15)]` , then sum of all the elements of matrix `A` is `0` b. `1` c. `2` d. `-3`

Options

A. 0
B. 1
C. 2
D. `-3`

Explanation

To solve the problem, we need to find the matrix \( A \) from the equation:

\[ \begin{pmatrix} 2 & -1 \\ 1 & 0 \\ -3 & 4 \end{pmatrix} A = \begin{pmatrix} -1 & -8 & -10 \\ 1 & -2 & -5 \\ 9 & 22 & 15 \end{pmatrix} \]

### Step 1: Define Matrix A
Let matrix \( A \) be defined as:

\[ A = \begin{pmatrix} a & b & c \\ d & e & f \end{pmatrix} \]

### Step 2: Perform Matrix Multiplication
Now we perform the multiplication of the two matrices:

\[ \begin{pmatrix} 2 & -1 \\ 1 & 0 \\ -3 & 4 \end{pmatrix} \begin{pmatrix} a & b & c \\ d & e & f \end{pmatrix} = \begin{pmatrix} 2a - d & 2b - e & 2c - f \\ a & b & c \\ -3a + 4d & -3b + 4e & -3c + 4f \end{pmatrix} \]

### Step 3: Set Up Equations
Now we equate the resulting matrix to the given matrix:

\[ \begin{pmatrix} 2a - d & 2b - e & 2c - f \\ a & b & c \\ -3a + 4d & -3b + 4e & -3c + 4f \end{pmatrix} = \begin{pmatrix} -1 & -8 & -10 \\ 1 & -2 & -5 \\ 9 & 22 & 15 \end{pmatrix} \]

From this, we can derive the following equations:

1. \( 2a - d = -1 \)
2. \( 2b - e = -8 \)
3. \( 2c - f = -10 \)
4. \( a = 1 \)
5. \( b = -2 \)
6. \( c = -5 \)
7. \( -3a + 4d = 9 \)
8. \( -3b + 4e = 22 \)
9. \( -3c + 4f = 15 \)

### Step 4: Solve for Variables
From equations 4, 5, and 6, we have:

- \( a = 1 \)
- \( b = -2 \)
- \( c = -5 \)

Now, substituting \( a \) into equation 1:

\[ 2(1) - d = -1 \implies 2 - d = -1 \implies d = 3 \]

Substituting \( b \) into equation 2:

\[ 2(-2) - e = -8 \implies -4 - e = -8 \implies e = 4 \]

Substituting \( c \) into equation 3:

\[ 2(-5) - f = -10 \implies -10 - f = -10 \implies f = 0 \]

### Step 5: Collect Values
Now we have:

\[ A = \begin{pmatrix} 1 & -2 & -5 \\ 3 & 4 & 0 \end{pmatrix} \]

### Step 6: Calculate the Sum of All Elements
Now we find the sum of all elements in matrix \( A \):

\[ 1 + (-2) + (-5) + 3 + 4 + 0 = 1 \]

### Final Answer
The sum of all elements of matrix \( A \) is \( 1 \).