Precalculus Matrix Systems Quiz: 10 Questions

Contents

Key takeaway: This precalculus matrix systems quiz has 10 multiple-choice questions on determinants, inverses, \(AX=B\), transformations and area. Some items include a graph or a parallelogram, so inspect the figure first. The topics line up with A-REI.C.8 and N-VM.C.12. Each explanation reveals the common slip behind the tempting wrong answers. A comfortable pace is 15 minutes in total.

Ten quick multiple-choice questions on determinants, inverses, the solution \(X=A^{-1}B\), singular systems and the effect of a matrix on the plane. Choose one answer each, then read the explanation to find which common slip each wrong option stands for.

PrecalculusCommon Core: A-REI.C.8 · A-REI.C.9 · N-VM.C.11 · N-VM.C.12

🎯 Solving Systems with Matrices – Precalculus math quiz

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Question 1

Find the determinant of the matrix with rows \(3,\ 4\) and \(2,\ 5\).

A \(23\)

B \(-7\)

C \(14\)

D \(7\)

Question 2

Which matrix is the inverse of the matrix with rows \(5,\ 2\) and \(7,\ 3\)?

A rows \(3,\ 2\) and \(7,\ 5\)

B rows \(-3,\ 2\) and \(7,\ -5\)

C rows \(3,\ -2\) and \(-7,\ 5\)

D rows \(5,\ -2\) and \(-7,\ 3\)

Question 3

A square matrix has determinant \(0\). What does this tell you?

A The matrix is the identity

B The system always has exactly one solution

C All entries of the matrix are \(0\)

D The matrix has no inverse

Question 4

For the matrix equation \(AX=B\) with \(A\) invertible, which expression gives \(X\)?

A \(BA^{-1}\)

B \(A^{-1}B\)

C \(AB^{-1}\)

D \(B-A\)

Question 5

The graph shows the system \(x+y=5\), \(x-y=1\). Which pair is the solution?

Two lines of a system.

A \((2,\ 3)\)

B \((3,\ 2)\)

C \((1,\ 4)\)

D \((3,\ -2)\)

Question 6

A matrix with rows \(2,\ 0\) and \(1,\ 3\) maps the unit square to the orange parallelogram. What is its area?

Image of the unit square.

A \(6\)

B \(5\)

C \(3\)

D \(7\)

Question 7

The matrix with rows \(1,\ 2\) and \(0,\ 3\) acts on the point \((2,\ -3)\). What is the image?

A \((-4,\ -9)\)

B \((2,\ -5)\)

C \((-4,\ 9)\)

D \((8,\ -9)\)

Question 8

A matrix has determinant \(-3\). It transforms a region of area \(5\). What is the area of the image?

A \(-15\)

B \(\frac{5}{3}\)

C \(2\)

D \(15\)

Question 9

A matrix keeps \((1,\ 0)\) fixed and sends \((0,\ 1)\) to \((0,\ -1)\). Which matrix is it?

Where the two basis points land.

A rows \(1,\ 0\) and \(0,\ -1\)

B rows \(-1,\ 0\) and \(0,\ 1\)

C rows \(1,\ 0\) and \(0,\ 1\)

D rows \(0,\ 1\) and \(1,\ 0\)

Question 10

The system \(x+2y=4\), \(3x+6y=12\) has \(\det A=0\). How many solutions does it have?

A Exactly one

B No solution

C Infinitely many

D Exactly two

🎉 Quiz complete!

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Remember:

A system of linear equations can be written as \(AX=B\).
For a \(2\times 2\) matrix, \(\det A=ad-bc\) and \(A^{-1}=\frac{1}{ad-bc}\) times the matrix with rows \(d,\ -b\) and \(-c,\ a\).
If \(\det A\neq 0\), then \(X=A^{-1}B\) is the only solution; if \(\det A=0\), there is no inverse.
The columns of a \(2\times 2\) matrix are the images of \((1,\ 0)\) and \((0,\ 1)\).
A transformation by \(M\) multiplies every area by \(|\det M|\).

Which questions check determinants and inverses?

The matrix with rows \(3,4\) and \(2,5\) has determinant \(15-8=7\) in the first item. Question 2 asks for the inverse of rows \(5,2\) and \(7,3\), which is rows \(3,-2\) and \(-7,5\). A determinant of 0 in question 3 means no inverse. Remember to swap the diagonal and negate the other two entries.

How do the solving questions work?

For \(AX=B\) in question 4, the answer is \(A^{-1}B\), with the inverse on the left. Question 5 shows two lines for \(x+y=5\) and \(x-y=1\), and the crossing point is \((3,2)\). Question 10 gives \(x+2y=4\) and \(3x+6y=12\), where the determinant is zero and the equations match. As a result, infinitely many solutions exist.

What about transformations and area?

A matrix with rows \(2,0\) and \(1,3\) maps the unit square to a parallelogram of area 6 in question 6. In question 7, rows \(1,2\) and \(0,3\) send \((2,-3)\) to \((-4,-9)\). Question 8 multiplies area 5 by \(|-3|\) to get 15. Finally, question 9 reads the matrix from the images of \((1,0)\) and \((0,1)\), and the columns are those images.

What is the best next step?

Compare your score with the topics: determinants, solving or geometry. Reread the matching lesson section for the weakest topic. Retake the quiz in a day, and then decide whether the unit test is ready for you.