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

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Grade 12 Matrices CCSS: HSN-VM.C.6

Matrices can be used to send coded messages. One way this is done is by taking a message, changing the letters into numbers, and then putting these numbers into a matrix. Zero represents a blank space. The matrix is formed by starting with the upper left most element, and going down each column till its full, then starting on the next column. If there are extra spaces in a matrix, fill the remaining elements with zeros (for example, if a message only has 7 letters, and one uses a 3-by-3 matrix, the remaining 2 elements would be zeros). Call the matrix containing the message matrix M. Then, one multiplies this by an encoding matrix; call this matrix A. Thus, the message sent would be the matrix AM. The recipient knows the encoding matrix, A. All they have to do is to multiply by the inverse,

$A^{-1}$ .

For the coded message

$[[24,20,56,76,55,24,20,56,37],[87,78,206,267,185,87,78,206,127],[27,95,173,118,-110,27,95,173,1]]$ , and the coding matrix

$[[-1,3,4],[-3,10,14],[2,-6,7]]$ , what is the message?

KEEP THIS TO YOURSELF ONLY
KEEP IT SECRET KEEP IT SAFE
DO NOT TELL ANYONE ANYTHING
KEEP CONCEALED AT ALL COSTS

Grade 12 Matrices CCSS: HSN-VM.C.6

Which of the following matrix equations correctly represents this system of equations?

$4x + 3y - z = 12, \ \ 3y+x+z= 10, \ \ 8x - z - 3y = 1$

$[[4,3,-1],[3,1,1],[8,-1,-3]] [[x],[y],[z]] = [[12],[10],[1]]$
$[[4,3,-1],[3,1,1],[8,-1,-3]] [[12],[10],[1]] = [[x],[y],[z]]$
$[[4,3,-1],[1,3,1],[8,-3,-1]] [[x],[y],[z]] = [[12],[10],[1]]$
$[[4,1,8],[3,3,-3],[-1,1,-1]] [[x],[y],[z]] = [[12],[10],[1]]$

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

Find the inverse of the matrix

$[[0,6],[2,8]]$ .

$[[4/6,2/4],[2/12,6/0]]$
$[[-2/3,1/2],[1/6,0]]$
$[[0,1/2],[1/6,2/3]]$
$[[96,-72],[-24,0]]$

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

Which matrix would transform the vector

$<< 4,2,7 >>$ to the vector

$<<20,23,51 >>$ ?

$[[0,3,2],[4,0,1],[3,2,5]]$
$[[3,2,5],[4,0,1],[0,3,2]]$
$[[0,4,3],[3,0,2],[2,1,5]]$
$[[5,1,2],[2,0,3],[3,4,0]]$

Grade 12 Matrices CCSS: HSN-VM.C.6

Which augmented matrix represents the system of equations

$2x+5y=32$ and

$4x-3y=14$ ?

$[[-2,-5,, -32],[-4,3,, -14]]$
$[[2,5,,32],[4,-3,,14]]$
$[[32,5,,2],[4,-3,,14]]$
$[[2,32,,5],[14,-3,,4]]$

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

For the vector

$<<3,-1,4>>$ , find the resulting vector if the transformation matrix

$[[3,2,1],[3,2,1],[3,2,1]]$ is applied to it.

$[[11],[11],[11]]$
$[[18],[12],[6]]$
$[[13],[13],[13]]$
This vector and matrix cannot be multiplied together.

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

Which one of the following matrix equations correctly shows the transformation of the first rectangle, whose vertex matrix is

$V_1 = [[1,1,4,4],[1,2,2,1]]$ , to the second rectangle, whose vertex matrix is

$V_2 = [[-4,-4,2,2],[1,3,3,1]] ?$
1x3 Q1 Horizontal

$V_2 = [[2,0],[0,2]] (V_1 - [[1,1,1,1],[1,1,1,1]] ) - [[4,4,4,4],[-1,-1,-1,-1]]$
$V_2 = [[2,0],[0,2]] V_1 - [[5,5,5,5],[0,0,0,0]]$
$V_2 = [[-4,1/2],[3/2, 1]] V_1$
$V_2 = [[2,2],[2,2]] ( V_1 - [[1,1,1,1],[1,1,1,1]])$

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

This question is a part of a group with common instructions. View group »

How does the absolute value of the determinant of the transformation matrix relate to the area of the two quadrilaterals (the square,

$S_1$ and the transformed shape,

$S_2$ )? Choose the equation which correctly describes this relationship.

$|det(T)| = "Area"(S_1) \ "Area"(S_2)$
$"Area"(S_2) = |det(T)| \ "Area"(S_1)$
$|det(T)| = |"Area"(S_2) - "Area"(S_1)|$
There is no relationship between the absolute value of the determinant of T and the area of the shapes.

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

In Leane's math homework, she has to multiply three matrices together by hand, as follows:

$[[8,2,0],[-4,7,-3]] * [[4,7,-4,-2],[1,3,8,7],[-5,-3,6,6]] * [[-6],[5],[-4],[1]]$ . She decides that if she multiplies the last two matrices together first, it will make the computation easier. Is she correct? Why or why not.

No, she cannot do this. She has to multiply the first two matrices together first, and then the third, according to normal evaluating rules.
Although she is allowed to do this, it will not make the computation easier. Regardless of how she multiplies these matrices, the answer will be a 2-by-1 matrix, and the same work will be involved either way.
Yes, she can do this, and it will make the computation easier. Multiplying the second and third matrices together first, and then the first by the product just found, will reduce the total number of arithmetic computations (multiplication and addition) to be performed.
These matrices are not able to be multiplied together, as they have incompatible dimensions.

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

Adrian was learning about the identity matrix and multiplicative inverses in math class, and is now working on his homework. He knows that, for a square matrix

$A$ , he can usually find another matrix

$B$ such that

$AB = I$ , where

$I$ is the identity matrix of the same size as

$A$ and

$B$ . Adrian is now considering the matrix

$A = [[3,2],[6,4]]$ . The way he has been finding the multiplicative inverse matrix is to create another matrix

$B = [[a,b],[c,d]]$ . He performs this multiplication, and it yields two systems of equations, as follows.

${{:(3a + 2c , = 1),(6a + 4c, = 0):}$

${{:(3b + 2d , = 0),(6b + 4d, = 1):}$

Adrian realizes that both these systems are inconsistent. Is Adrian's work correct? And if so, what does this imply?

No, Adrian's work is not correct. His entire method for finding multiplicative inverses is wrong.
No, Adrian's work is not correct. Although his general method is OK, he has made a mistake when multiplying the matrices.
Yes, Adrian's work is correct. This means that, in this case, no such B matrix exists, and therefore not all square matrices have a multiplicative inverse.
Yes, Adrian's work is correct. This implies that, in this case, the B matrix must be the zero matrix.

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

Can the following matrix expression be evaluated? Why or why not?

$[[3,-4],[0,-1],[9,1]] \ * ( [[0,1],[1,0]] + [[-5,-9],[-1,4]] )$

Yes, but only as is (one cannot distribute the 3-by-2 matrix).
Yes, either as is, or if the distributive property is applied.
No, after performing the addition of the two smaller matrices, the new dimensions will not allow for multiplication with the left-most matrix.
No, all matrices must be have the same dimensions.

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

Given the matrix multiplication equation

$A \ B = C$ , where

$C = [[4,3,2,0],[-1,3,5,9]]$ , which of the following are possible matrices for B? There may be more than one correct choice.

$B = [[1,0,0,0],[0,0,0,1]]$
$B = [[13,-6,14,14],[0,-6,8,3],[-10,12,1,0],[-3,-6,4,11],[2,15,13,8]]$
$B = [[3,-2,1,0],[9,3,-11,13],[2,4,-5,-6],[-1,-2,10,6]]$
$B = [[4,-1],[2,-3],[-2,1],[0,5]]$

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

Multiply, if possible.

$[[3,2,-1],[0,9,1],[3,2,-5]] * [[1,0,1],[-2,-3,2],[5,6,-1]]$

$[[3,0,-1],[0,-27,2],[15,12,5]]$
$[[2,-14,28],[1,-25,53],[-2,-22,32]]$
$[[-5,-12,9],[-23,-22,18],[-25,-37,16]]$
$[[-6,-12,8],[-13,-21,17],[-26,-36,12]]$

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

Shane has a matrix of three ordered pairs,

$P = [[3,2],[8,-1],[5,0]]$ , where the x-coordinates are in the first column and the y-coordinates are in the second column. He wants to find a matrix A, such that the product

$A \ P$ gives a matrix that will only have the first and third ordered pairs. Which of the following matrices will give this result?

$A = [[1,1],[0,0],[1,1]]$
$A = [[1,0,0],[0,0,1]]$
$A = [[1,0,1]]$
$A = [[1,0,1],[1,0,1]]$

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

Multiply.

$[[2,-1],[0,7]] * [[-1,-4],[6,1]]$

$[[-8,-9],[42,7]]$
$[[-2,4],[0,7]]$
$[[1,-5],[6,8]]$
$[[-8,-27],[12,1]]$

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

Find the inverse of the matrix

$[[2,5],[1,6]]$ .

$[[42,-35],[-7,14]]$
$[[6/17,-5/17],[-1/17,2/17]]$
$[[2/7,5/7],[1/7,6/7]]$
$[[6/7,-5/7],[-1/7,2/7]]$

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

Multiply the following.

$[(0,2),(-2,-5)] * [(6,-6),(3,0)]$

Undefined
$[(0,6),(-27,12)]$
$[(6,0),(-27,12)]$
None of the above

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

Put the vertices of the shape in the graph below into a vertex matrix, and then apply the rotation matrix

$A = [[sqrt(2)/2, -sqrt(2)/2],[sqrt(2)/2, sqrt(2)/2]]$ . By what angle, measured in degrees, has the shape been rotated (given that the angle of rotation is counterclockwise)?

30°
45°
60°
105°

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

For the matrix

$A = [[1,0,0],[0,1,0],[0,0,0]]$ , what is the best description of how this transforms a vector with 3 components if they are multiplied together?

The vector stays the same.
The vector now only has 2 components.
The vector's third component is changed to zero.
At least one component of the vector is reduced to zero.

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

Given the triangle with vertices

$(1,1), (2,3),$ and

$(5,1)$ , which of the following matrix expressions would represent the reflection of this triangle over the y-axis?

$[[1,0],[0,-1]] \ [[1,1],[2,3],[5,1]]$
$[[-1,0],[0,1]] \ [[1,2,5],[1,3,1]]$
$[[-1,0],[0,1]] \ [[1,1],[2,3],[5,1]]$
$[[1,0],[0,-1]] \ [[1,2,5],[1,3,1]]$

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Twelfth Grade (Grade 12) Matrices Questions