Question

Consider the following matrices.
A =
2 4 1 −3
and B =
1 −2 0 5
Write the given matrix as a linear combination of A and B in M2, 2. (If an answer does not exist, enter DNE.)
4 16 3 −19
=
2 4 1 −3
+
1 −2 0 5

Answer 1

Answer:

  3A -2B

Step-by-step explanation:

You want a linear combination of matrices A[2, 4, 1, -3] and B[1, -2, 0, 5] that gives the matrix [4, 16, 3, -19].

Linear combination

The desired linear combination applies to every triple of corresponding elements, so we can write two linear equations in the two unknown coefficients. If the result matrix is C, we have ...

  xA +yB = C

Using the first and last elements of the matrices involved, we get the equations ...

• 2x +y = 4
• -3x +5y = -19

Solving these equations by a convenient method gives (x, y) = (3, -2).

The desired linear combination is ...

  3A -2B = C

__

Additional comment

The first attachment shows the solution for x and y. The second attachment verifies that it gives the desired result matrix.

We could write the matrices involved as 2×2 square matrices. For the purpose of the required relationship, the dimensions of the matrix are immaterial. The relationship applies on an element-by-element basis.

<95141404393>