This precalculus matrix systems worksheet gives 24 problems on solving systems with matrices, ordered from easy to hard, with a separate answer key.
These 24 problems build the matrix toolkit step by step: writing a system as \(AX=B\), computing determinants and inverses, solving systems with \(X=A^{-1}B\), and using matrices as transformations of the plane (A-REI.C.8, A-REI.C.9, N-VM.C.11, N-VM.C.12).
The last six are word problems, including a three-item bake sale that needs technology. Every problem has a worked solution in the answer key.
Precalculus Matrix Systems Worksheet: problems 1–5
Problem 1: Write a system as AX = B
Write the system \(3x + y=7\), \(2x – y=3\) as a matrix equation \(AX=B\). Name the matrices \(A\), \(X\) and \(B\).
Problem 2: Read a matrix equation
Matrix \(A\) has rows \(4,\ -1\) and \(2,\ 5\). Write the equations of \(AX=B\) when \(B=(9,\ -1)\), then verify that \((2,\ -1)\) is a solution.
Problem 3: Determinants of 2 by 2 matrices
Find the determinant of each matrix. Each is written as its rows.
- rows \(3,\ 5\) and \(2,\ 4\)
- rows \(-2,\ 6\) and \(1,\ -3\)
- rows \(4,\ -1\) and \(3,\ 2\)
- rows \(\frac{1}{2},\ 3\) and \(2,\ 8\)
Problem 4: Does the inverse exist?
Decide whether each matrix has an inverse. Explain using the determinant.
- rows \(6,\ 9\) and \(2,\ 3\)
- rows \(5,\ 2\) and \(3,\ 1\)
- rows \(-4,\ 8\) and \(1,\ -2\)
- rows \(7,\ 1\) and \(0,\ 3\)
Problem 5: Inverse with determinant 1
Find the inverse of the matrix with rows \(3,\ 5\) and \(1,\ 2\).
Precalculus Matrix Systems Worksheet: problems 6–10
Problem 6: Inverses and the identity
Find the inverse of each matrix (rows listed) and show that \(AA^{-1}=I\) for part (a).
- rows \(4,\ 3\) and \(2,\ 2\)
- rows \(5,\ 2\) and \(-3,\ 4\)
- rows \(1,\ 2\) and \(3,\ 5\)
Problem 7: Are these inverses?
Matrix \(A\) has rows \(2,\ 3\) and \(1,\ 2\). Matrix \(C\) has rows \(2,\ -3\) and \(-1,\ 2\). Multiply \(AC\) and \(CA\) to decide whether \(C=A^{-1}\).
Problem 8: Solve with a given inverse
The inverse of the matrix with rows \(2,\ 5\) and \(1,\ 3\) has rows \(3,\ -5\) and \(-1,\ 2\). Use it to solve \(2x+5y=7\), \(x+3y=4\).
Problem 9: Solve a system with the inverse
Solve \(3x+4y=10\), \(2x+3y=7\) by writing \(AX=B\) and using \(A^{-1}\).
Problem 10: Determinant 11
Solve \(5x-2y=4\), \(3x+y=9\) with the inverse matrix.
Precalculus Matrix Systems Worksheet: problems 11–15
Problem 11: A fractional solution
Solve \(4x+y=3\), \(6x+2y=5\) with an inverse matrix.
Problem 12: When the determinant is zero
For each system, find \(\det A\), then decide whether the system has one solution, none or infinitely many.
- \(x+2y=3\), \(2x+4y=6\)
- \(x+2y=3\), \(2x+4y=7\)
Problem 13: Apply a matrix to a point
Matrix \(P\) has rows \(2,\ 0\) and \(0,\ 3\). Matrix \(R\) has rows \(0,\ -1\) and \(1,\ 0\). Find
- \(P\) times \((4,\ -1)\)
- \(R\) times \((2,\ 5)\)
- \(R\) times \((-3,\ 1)\)
Problem 14: Shear the unit square
A shear matrix \(S\) has rows \(1,\ 2\) and \(0,\ 1\). Find the images of the vertices \((0,\ 0)\), \((1,\ 0)\), \((1,\ 1)\), \((0,\ 1)\) of the unit square shown, and sketch the image.
Problem 15: Build the matrix from two images
A matrix sends \((1,\ 0)\) to \((0,\ 1)\) and \((0,\ 1)\) to \((-1,\ 0)\), as shown. Write the matrix, then find the image of \((3,\ 2)\).
Precalculus Matrix Systems Worksheet: problems 16–20
Problem 16: Reflections
Matrix \(T\) has rows \(1,\ 0\) and \(0,\ -1\). The point \((5,\ 4)\) is plotted.
- Find the image of \((5,\ 4)\).
- Describe the transformation.
- Find \(\det T\) and explain what the sign tells you.
Problem 17: Area scale factor
Let \(A\) have rows \(3,\ 1\) and \(2,\ 4\).
- Find the area scale factor.
- A triangle has area \(6\). What is the area of its image?
- A region maps to an image of area \(35\). What was its area?
Problem 18: Negative determinant
Matrix \(B\) has rows \(1,\ 2\) and \(3,\ 1\). A triangle of area \(4\) is transformed by \(B\). Find the area of the image and state what the sign of \(\det B\) means.
Problem 19: School store prices
At the school store, Ethan buys 3 pens and 2 notebooks for \(\$7.25\). Maya buys 2 pens and 5 notebooks for \(\$11.25\). Write a matrix equation and use the inverse to find the price of one pen and one notebook.
Problem 20: Farmers market
At a farmers market, 4 pounds of apples and 3 pounds of pears cost \(\$19.00\). Also, 3 pounds of apples and 2 pounds of pears cost \(\$13.50\). Use \(X=A^{-1}B\) to find each price per pound.
Precalculus Matrix Systems Worksheet: problems 21–24
Problem 21: Bake sale with three items
At a bake sale, Priya sells cookies, brownies and muffins at three prices. Day 1: 4 cookies, 3 brownies and 2 muffins earn \(\$17.00\). Day 2: 2 cookies, 5 brownies and 1 muffin earn \(\$15.50\). Day 3: 1 cookie, 2 brownies and 6 muffins earn \(\$20.50\). Write \(AX=B\) and use technology to find \(A^{-1}\) and each price.
Problem 22: Scaling a game sprite
A game designer enlarges a rectangular sprite measuring 5 units by 4 units with the matrix with rows \(3,\ 0\) and \(0,\ 2\). Find the new dimensions and show that the area changes by \(|\det|\).
Problem 23: Mapping a triangular lot
A surveyor transforms a triangular lot with vertices \((0,\ 0)\), \((4,\ 0)\) and \((0,\ 3)\) using the matrix with rows \(2,\ 1\) and \(-1,\ 3\). Find the image vertices. Then find the area of the image two ways.
Problem 24: Undo a transformation
Matrix \(H\) has rows \(2,\ 1\) and \(1,\ 1\). A point \(P\) is sent to \((11,\ 8)\) by \(H\). Find \(P\), and explain why the area of the figure is unchanged.
How to use this precalculus matrix systems worksheet
Print this precalculus matrix systems worksheet as a PDF, solve every problem on paper, then compare with the answer key.
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