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

If Matrix A =

$[(-3,1),(-2,4),(5,-1)]$ and Matrix B =

$[(4,-3),(0,-2),(-2,4)]$ , then what is 3A - 2B?

$[(-1,-3),(-6,8),(11,5)]$
$[(-1,9),(-6,8),(11,5)]$
$[(-1,9),(-6,8),(11,-11)]$
$[(-17,9),(-6,16),(19,-11)]$

Grade 11 Matrices CCSS: HSA-REI.C.8

When representing a system of linear equations in two variables as a matrix equation, one can use the general form

$A [[x],[y]] = vec[b]$ . For a given system of linear equations, is it possible for the 2-by-2 matrix

$A$ to have different entries? Why or why not?

Yes. Either equation in the system of equations can be multiplied by a constant. This will also affect $vec(b)$ .
No. A different matrix $A$ would result in a different answer regardless of anything else.
It depends on whether the system of equations is dependent or independent.
No. Only a system of equations that is inconsistent, and therefore has no answer anyways, can have different $A$ matrices.

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

The matrix

$[[1,2,0],[3,4,2],[3,1,5]]$ has an inverse.

True
False

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

For the matrices

$A = [[6,5],[0,0]]$ and

$B = [[0,0],[-4,2]]$ ,

$AB = BA$ .

True
False

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

If A is a given square matrix, and it is known that there exists a matrix B such that

$AB=1$ , which of the following would be the most efficient ways to find the matrix B?

Find the inverse of A. This is the matrix B.
Find the transpose of A. This is the matrix B.
Create a matrix B whose elements are variables. Then, perform matrix multiplication with the matrix A, setting each resulting entry equal to one. Solve this system of equations, which will give the elements of matrix B.
Multiply both sides of the equation, on the left, by slight variations of the matrix A. When one of these matrices, multiplied by A, becomes the identity matrix, this is the matrix B.

Grade 11 Matrices

State the dimensions of matrix

$F$ if

$F=[[0,1,0],[2,-4,2],[4,-8,4],[8,-16,8]]$ .

$16xx8$
$2xx2xx3$
$4xx3$
$3xx4$

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

What is the determinant of

$[[3,2],[6,4]]$ ?

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

What is the determinant of

$[[5,7,-2],[3,0,1],[-9,-7,2]] ?$

-112
-28
98
-98

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.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.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 11 Matrices

What is the determinant of the following matrix?

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

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

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