Common Core Standard HSN-VM.C.10 Questions

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

For

$M = [[6,-3],[-2,1]]$ , find

$|M|$ .

-9
16
12
0

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

Find the determinant of the following.

$[(-11,3,10),(19,2,-4),(-8,-10,20)]$

-2478
-2784
2784
none of the above

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

For

$M = [[4,-5],[-3,6]]$ , find

$M^{-1}$ , if it exists.

Matrix M does not have an inverse.
$[[2/13, 5/39],[1/13, 4/39]]$
$[[2/3, 5/9],[1/3, 4/9]]$
$[[4/9, -5/9],[-1/3, 2/3]]$

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

Evaluate the determinant using diagonals.

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

-80
-284
24
80

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

Find the inverse of the following matrix, if it exists.

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

There is no inverse of this matrix.
$[[-2/7, 10/7, -1/7],[-1/3, 4/3, 0],[1/7, -3/14, 1/14]]$
$[[2/7, -10/7, 1/7],[1/3, -4/3, 0],[-1/7, 3/4, -1/4]]$
$[[-2/7, -10/7, -1/7],[1/3, 4/3, 0],[1/7, 3/4, 1/4]]$

Grade 11 Matrices CCSS: HSN-VM.C.10

Find the inverse of

$A = [[4,6],[2,3]]$ , if it exists.

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

The matrix

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

True
False

Grade 12 Functions and Relations CCSS: HSN-VM.C.10

The zero matrix is always square.

True
False

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.10

The identity matrix is always square.

True
False

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

Matrix A is a square matrix and

$|A| = -3$ . With this information, which of the following has to be true about matrix A?

It is invertible.
It has more negative elements than positive elements.
It has an odd number of rows and columns.
One cannot take the transpose of it.

Grade 11 Matrices CCSS: HSN-VM.C.10, HSA-REI.C.9

Solve.

$[[5,4],[-2,-3]] [[x],[y]] = [[6],[-1]]$

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

$I$ is the identity matrix, and

$A$ has the same dimensions as

$I$ , then

$AI = IA$ .

True
False

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

Sarah has been given the following matrix equation to solve,

$A x = b$ , where A is a 3-by-3 matrix, b is a 3-by-1 matrix, and x is a 3-by-1 matrix. If she knows the equation can be solved by multiplying the inverse of A on both sides of the equation, which of the following must be true?

A must be a diagonal matrix (its off-diagonal entries must be zero).
The matrix b must also have an inverse.
The determinant of A is not equal to zero.
More than half of the elements in the matrix A are non-zero.

Grade 12 Functions and Relations CCSS: HSN-VM.C.10

For the 3-by-3 identity matrix

$I_3$ , and a given non-square matrix

$A$ , if

$I_3A = A$ , then

$AI_3=A$ .

True
False

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 Functions and Relations CCSS: HSN-VM.C.10

Which of the following are identity matrices? There may be more than one correct answer.

$[[1,0,0],[0,1,0],[0,0,1]]$
$[[1,0,0],[1,0,0],[1,0,0]]$
$[[1,0],[0,1]]$
$[[1,1],[1,1]]$

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Common Core Standard HSN-VM.C.10 Questions

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