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Type: Multiple-Choice
Category: Matrices
Level: Grade 12
Standards: HSN-VM.C.9
Author: nsharp1
Created: 5 years ago

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Grade 12 Matrices CCSS: HSN-VM.C.9

Simon was given a simple matrix equation, but some of the entries of the first matrix are missing. The equation is as follows.

$([[a,1],[-2,b]] \ [[5,2],[-3,5]] ) \ [[0,-1],[1,0]] = [[13,-17],[-9,7]]$

Simon's work to solve this is as follows:

$L.H.S. = ([[a,1],[-2,b]] \ [[5,2],[-3,5]] ) \ [[0,-1],[1,0]]$

$\ \ \ \ \ \ \ \ \ \ \ \ \ = [[a,1],[-2,b]] \ ([[5,2],[-3,5]] \ [[0,-1],[1,0]] )$

$\ \ \ \ \ \ \ \ \ \ \ \ \ = [[a,1],[-2,b]] \ [[2,-5],[5,3]]$

The first element of the first row will be

$2a+5$ and the first element of the second row will be

$-4+5b$ . Since these have to be equal to the elements of the matrix on the right hand side, this means that

$2a+5 = 13$ and

$-4+5b = -9$ . Solving these equations yields

$a = 4$ and

$b=-1$ .

Is Simon's work correct? If not, where did he make a mistake?

This is correct.
This is not correct. Simon cannot move the parenthesis as he did in his first step.
This is not correct. Simon made a mistake in multiplying the two matrices.
This is not correct. Simon only did part of the matrix multiplication to find the two equations. The full matrix multiplication will yield four equations, leading to an overdetermined system of equations that is inconsistent.