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# Common Core Standard HSN-VM.C.10 Questions

(+) Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.

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Matrix A is a square matrix and $|A| = -3$. With this information, which of the following has to be true about matrix A?
1. It is invertible.
2. It has more negative elements than positive elements.
3. It has an odd number of rows and columns.
4. One cannot take the transpose of it.
Grade 11 Matrices CCSS: HSN-VM.C.10, HSA-REI.C.9
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?
1. A must be a diagonal matrix (its off-diagonal entries must be zero).
2. The matrix b must also have an inverse.
3. The determinant of A is not equal to zero.
4. More than half of the elements in the matrix A are non-zero.
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?
1. Try at least 3 matrices, and if none of them multiplied by $A$ equal the identity matrix, then it is not possible.
2. Find the inverse of $A$, and then show that since this matrix is unique, there cannot exist another matrix $B$ such that $AB = I$.
3. 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.
4. 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. $[[1,0,0],[0,1,0],[0,0,1]]$
2. $[[1,0,0],[1,0,0],[1,0,0]]$
3. $[[1,0],[0,1]]$
4. $[[1,1],[1,1]]$
If $A$ is a 5-by-3 non-zero matrix, which of the following expressions, if evaluated, would yield the matrix $A$ as the answer? Choose all correct answers.
1. $A + [[0,0,0],[0,0,0],[0,0,0],[0,0,0],[0,0,0]]$
2. $[[0,0,0],[0,0,0],[0,0,0],[0,0,0],[0,0,0]] + A$
3. $A + [[0,0,0],[0,0,0],[0,0,0]]$
4. $A + [[0,0,0,0,0],[0,0,0,0,0],[0,0,0,0,0],[0,0,0,0,0],[0,0,0,0,0]]$
If $B$ is a 3-by-2 matrix, which of the following expressions, if evaluated, would yield the matrix $B$ as the answer? Choose all correct answers.
1. $B * [[1,0,0],[0,1,0],[0,0,1]]$
2. $B * [[1,0],[0,1]]$
3. $[[ 1,0,0],[0,1,0],[0,0,1]] * B$
4. $[[1,0],[0,1]] * B$
Given a matrix A, is there always another matrix, B, such that their sum is the zero matrix? What are the elements of the matrix B?
1. No, there isn't always such a matrix B. If there is, its elements are the negative of the corresponding elements in matrix A.
2. No, there isn't always such a matrix B. If there is, its elements are determined on a case-by-case basis, with no general rule.
3. Yes, there is always such a matrix. The elements of B are found on a case-by-case basis, with no general rule.
4. Yes, there is always such a matrix. The elements of B are the negative of the corresponding elements in A.
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?
1. Find the inverse of A. This is the matrix B.
2. Find the transpose of A. This is the matrix B.
3. 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.
4. 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.
For $M = [[4,-5],[-3,6]]$, find $M^{-1}$, if it exists.
2. $[[2/13, 5/39],[1/13, 4/39]]$
3. $[[2/3, 5/9],[1/3, 4/9]]$
4. $[[4/9, -5/9],[-1/3, 2/3]]$
Find the inverse of the following matrix, if it exists. $[[4,-3,8],[1,0,2],[-5,6,4]]$
2. $[[-2/7, 10/7, -1/7],[-1/3, 4/3, 0],[1/7, -3/14, 1/14]]$
3. $[[2/7, -10/7, 1/7],[1/3, -4/3, 0],[-1/7, 3/4, -1/4]]$
4. $[[-2/7, -10/7, -1/7],[1/3, 4/3, 0],[1/7, 3/4, 1/4]]$