Find a matrix A such that 2A - 3B + 5C = O, where B = [−2 2 0; 3 1 4] and C = [2 0 −2; 7 1 6].

To find the matrix A that satisfies the equation 2A - 3B + 5C = O, where B and C are given matrices, we can follow these steps:

Step 1: Define the Matrices

We have:

• B = [−2 2 0; 3 1 4]
• C = [2 0 −2; 7 1 6]

Step 2: Calculate 3B and 5C

First, we need to calculate 3B and 5C:

Calculating 3B

Multiply each element of B by 3:

• 3B = [−6 6 0; 9 3 12]

Calculating 5C

Now, multiply each element of C by 5:

• 5C = [10 0 −10; 35 5 30]

Step 3: Substitute and Rearrange

Now substitute these results into the equation:

2A - 3B + 5C = O becomes:

2A - [−6 6 0; 9 3 12] + [10 0 −10; 35 5 30] = O

Step 4: Combine the Matrices

Combine the matrices:

• Result = [−6 + 10 6 + 0 0 − 10; 9 + 35 3 + 5 12 + 30]

This simplifies to:

• Result = [4 6 -10; 44 8 42]

Step 5: Solve for A

Now, we have:

2A = Result

So:

A = Result / 2 = [2 3 -5; 22 4 21]

Final Answer

The matrix A is:

A = [2 3 -5; 22 4 21]