Twelfth Grade (Grade 12) Matrices Questions for Tests and Worksheets

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Grade 12 Matrices CCSS: HSN-VM.C.8

For given matrices A and B, let

$A \ B = M$ , where M is a also a matrix. Which of the following correctly describes the dimensions of matrix M?

Number of rows of A, number of columns of B.
Number of rows of B, number of columns of A.
Number of rows of A, number of columns of A.
Number of rows of B, number of columns of B.

Grade 12 Matrices CCSS: HSN-VM.C.8

Find the difference.

$[[2,0],[-4,1],[8,12],[0,-3],[1,1]] - [[-3,5],[6,10],[3,12],[-4,4],[8,7]]$

$[[5,5],[-2,-9],[-5,-24],[-4,1],[-7,-6]]$
$[[5,-5],[-10,-9],[5,0],[4,-7],[-7,-6]]$
$[[-1,5],[2,11],[11,24],[-4,1],[9,8]]$
$[[-5,-5],[-2,9],[5,24],[-4,-7],[7,6]]$

Grade 12 Matrices CCSS: HSA-REI.C.9

Find the inverse of the matrix

$[[1,2],[3,4]]$ .

$[[-2,1],[3/2,-1/2]]$
$[[4,-2],[-3,1]]$
$[[4,2],[3,1]]$
$[[1,-2],[-3,4]]$

Grade 12 Matrices CCSS: HSN-VM.C.10

James needs to show that for matrix

$A = [[5,-2,1],[3,3,2],[-6,3,-1]]$ , there is no matrix

$B, B!=I$ , such that

$AB = I$ , where

$I$ is the 3-by-3 identity matrix. How can he do this?

Try at least 3 matrices, and if none of them multiplied by $A$ equal the identity matrix, then it is not possible.
Find the inverse of $A$ , and then show that since this matrix is unique, there cannot exist another matrix $B$ such that $AB = I$ .
Subtract by the additive inverse on both sides, and then factor the left hand side of the equation. This implies that if $B=I$ the equation equals the zero matrix, which it can't.
Show that the determinant of $A$ is zero, which means that it does not have a multiplicative inverse.

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.

Grade 12 Matrices CCSS: HSN-VM.C.10

What is the determinant of

$[[1,-3,4],[6,2,-1],[0,3,5]]$ ?

Grade 12 Matrices CCSS: HSN-VM.C.7

Matrix B is given with two of its elements unknown,

$B = [[B_(1,1), 0],[-4, B_(2,2)]]$ . Another matrix is given, which is matrix B which has been multiplied by a scalar,

$[[3,0],[-1,2]]$ . What are the two unknown elements in matrix B, listed as

$B_(1,1), B_(2,2) ?$

$12, 8$
$3/4, 1/2$
$-12, -8$
$-3/4, -1/2$

Grade 12 Matrices CCSS: HSN-VM.C.7

$G = [[4,5],[-2,6]]$ and

$a G = [[8,10],[-4,12]]$ , what is the value of

$a ?$

-1/2
1/2
-2
2

Grade 12 Matrices CCSS: HSN-VM.C.8

Add.

$[[4,9,-6],[-3,0,8],[2,2,5]] + [[-10, 2,6],[5,-7,-8],[2,12,-5]]$

$[[-6,11,0],[2,-7,0],[4,14,0]]$
$[[14,11,12],[8,-7,16],[4,14,10]]$
$[[-14, 7,-12],[-8,7,16],[0,-10,10]]$
$[[-6,11],[2,-7],[4,14]]$

Grade 12 Matrices CCSS: HSN-VM.C.8

Perform the indicated operations. If the matrix does not exist, choose impossible.

$[[8,3],[-1,-1]]-[[0,-7],[6,2]]$

$[[-8,-10],[-7,-3]]$
$[[-3,10],[-7,8]]$
$"Impossible"$
$[[8,10],[-7,-3]]$

Grade 12 Matrices CCSS: HSN-VM.C.10

The 5x5 identity matrix of looks like which of the following?

$[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]$
$[[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]$
$[[0,0,1,0,0],[0,0,1,0,0],[1,1,1,1,1],[0,0,1,0,0],[0,0,1,0,0]]$
A and B

Grade 12 Matrices CCSS: HSN-VM.C.11

The matrix

$[[1,2,3],[4,5,6],[7,8,9]]$ multiplied by the vector

$<< 1,2,3 >>$ equals which of the following?

$[[1,4,9],[4,10,18],[7,16,27]]$
$[ [14],[32],[50]]$
$[[12],[30],[54]]$
$[[1,2,3],[8,10,12],[21,24,27]]$

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

The vertex matrix for rectangle ABCD is

$V = [[1,1,4,4],[1,3,3,1]]$ . Which of the following is the correct transformed vertex matrix, if the transformation matrix

$A = [[0,1],[-1,0]]$ is applied?

$[[1,-1],[3,-1],[3,-4],[1,-4]]$
$[[-1,-3,-3,-1],[1,1,4,4]]$
$[[1,3,3,1],[-1,-1,-4,-4]]$
$[[-1,-3,-3,-1],[-1,-1,-4,-4]]$

Grade 12 Matrices CCSS: HSN-VM.C.11

For the transformation matrix

$A = [[1,0],[0,-1]]$ , how is the vector

$vec{v} = <<3, 4>>$ affected if it is multiplied with

$A ?$

It has been rotated 90° clockwise.
It has reversed direction.
It has been reflected over a horizontal line.
The resulting transformation is a combination of reflections and rotations.

Grade 12 Matrices CCSS: HSN-VM.C.10

The matrix

$[[5,-1],[-10,2]]$ has an inverse.

True
False

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

For matrices A, B, C, D, which of the following is always equivalent to the expression

$A(B+C)D ?$ There may be more than one correct answer.

$A D (B+C)$
$ABD + ACD$
$(AB +AC)D$
$A (BD + CD)$

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

Jennie is talking with Liz after their math class. Jennie says, from what she understood from the class, that for matrices

$A$ and

$B$ ,

$AB$ is never equal to

$BA$ . Liz thinks that, although this is true in general, there are certain matrices for which

$AB = BA$ . Who is correct, and why?

Jennie is correct. Since the distributive property does not apply, $AB != BA$ , for all matrices.
Liz is correct. One simply has to take the transpose of the two matrices (the rows become columns of each matrix), and then it is usually true that $AB = BA$ .
Liz is correct. For example, the identity matrix with any square matrix of the same size.
Neither is correct. $AB = BA$ is always true for any matrices A, B.

Grade 12 Matrices CCSS: HSN-VM.C.7

If the matrix

$[[2,9,8],[0,3,4],[1,11,3]]$ is multiplied by the scalar 3, what is the result?

$[[2/3,3,8/3],[0,1,4/3],[1/3,11/3,1]]$
$[[6,0,3],[27,9,33],[24,16,12]]$
$[[5,12,11],[3,6,7],[4,14,7]]$
$[[6,27,24],[0,9,12],[3,33,9]]$

Grade 12 Matrices CCSS: HSN-VM.C.7

If the matrix

$[[6,8,9],[1,0,2],[3,6,2]]$ is multiplied by the scalar 3, what is the result?

$[[9,11,12],[4,3,5],[8,9,5]]$
$[[18,24,27],[3,0,6],[9,18,6]]$
$[[3,5,6],[-2,-3,-1],[0,3,-1]]$
$[[9,8,6],[2,0,1],[2,6,3]]$

Grade 12 Matrices CCSS: HSA-REI.C.9

Find the inverse of the matrix

$[[10,3],[2,4]]$ .

$[[5/17,3/34],[1/17,2/17]]$
$[[2/23,-3/46],[-1/23,5/23]]$
$[[2/17,-3/34],[-1/17,5/17]]$
$[[4,-3],[-2,10]]$

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Twelfth Grade (Grade 12) Matrices Questions