Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
The last problems bring in a food truck, a chemistry mixture, a 3 × 3 ticket system and a robot work zone.
Standards: A-REI.C.8, A-REI.C.9, N-VM.C.11, N-VM.C.12 · Level: Grade 12 Precalculus
Problem 1 Writing a Matrix Equation ★★★
Write the system 4x − y = 9 and 2x + 5y = −3 in the form AX = B. Name A, X and B.
Problem 2 Reading a Matrix Equation ★★★
Write the matrix equation as two equations, then check whether (1, −2) is a solution. A has rows (2, −3) and (5, 1), X is the column (x, y), and B is the column (8, 3).
Problem 3 Is There One Solution? ★★★
Decide whether the system 6x + 9y = 5 and 2x + 3y = 7 has a unique solution. Use the determinant.
Problem 4 Matrix Times a Point ★★★
Find the product of the matrix with rows (3, −1) and (2, 4) and the column (5, −2).
Problem 5 Reflecting a Point ★★★
The matrix with rows (−1, 0) and (0, 1) reflects points in the y-axis. Find the image of (6, −4).
Problem 6 Stretching the Unit Square ★★★
The matrix with rows (3, 0) and (0, 2) transforms the unit square into the rectangle shown. Find its determinant and the area of the image.
Problem 7 Solving with an Inverse ★★★
Solve 4x + 3y = 6 and 5x + 2y = 11 using an inverse matrix.
Problem 8 A Negative Solution ★★★
Solve 3x + 4y = −7 and 2x − y = −12 with matrices.
Problem 9 A Fractional Solution ★★★
Solve 2x + 3y = 5 and 4x − y = 6 using X = A⁻¹B.
Problem 10 Two Moves in a Row ★★★
Apply the rotation with rows (0, −1) and (1, 0) to the point (−3, 5), then reflect the result in the x-axis using the matrix with rows (1, 0) and (0, −1). Then find one matrix that does both moves at once.
Problem 11 Area After a Transformation ★★★
A triangle with area 6 square units is transformed by the matrix with rows (4, −2) and (1, 3). Find the area of the image.
Problem 12 Building a Scaling Matrix ★★★
Write the matrix that doubles every x-coordinate and triples every y-coordinate. Find the image of (2, −1) and the area of the image of the unit square.
Problem 13 A Matrix That Flattens ★★★
The matrix with rows (2, 4) and (3, 6) is applied to the plane. Find its determinant, the images of (1, 0) and (0, 1), and describe what happens to the plane.
Problem 14 A Shear ★★★
The matrix with rows (1, 2) and (0, 1) shears the triangle with vertices (0, 0), (3, 0) and (0, 2). Find the image of each vertex and compare the two areas.
Problem 15 Food Truck Prices ★★★
A food truck sold 18 wraps and 12 bowls on Friday for $246, and 25 wraps and 20 bowls on Saturday for $365. Set up a matrix equation and use an inverse to find the price of a wrap and of a bowl.
Problem 16 Mixing Solutions ★★★
A chemistry club mixes a 20% acid solution with a 50% acid solution to make 12 liters of 38% acid solution. Let x and y be the liters of each. Write a matrix equation and solve it with an inverse.
Problem 17 Theater Ticket Prices ★★★
A theater sells adult, child and senior tickets at prices a, c and s. Three shows gave these totals: Show 1 sold 40 adult, 30 child and 10 senior tickets for $810. Show 2 sold 25, 45 and 20 for $840. Show 3 sold 30, 20 and 30 for $790. Write the system as AX = B (divide each equation by 5 first) and use technology to find the three prices.
Problem 18 Robot Arm Zone ★★★
A robot’s work zone is the triangle with vertices (2, 1), (6, 1) and (2, 4). A controller applies the matrix with rows (0, −2.5) and (2.5, 0), which rotates by 90° counterclockwise and scales by 2.5. Find the image of each vertex and the area of the new zone two ways.
Problem 19 Transformation from Images ★★★
A transformation sends (1, 0) to (3, 1) and (0, 1) to (−1, 2). (a) Write its matrix. (b) Find the image of (2, 5). (c) Find the determinant and say how areas change. (d) Find the point that is sent to (8, 5).
Problem 20 When Is There No Unique Solution? ★★★
Consider 2x + ky = 8 and 3x + 6y = 12. (a) Find the value of k for which the system has no unique solution. (b) For that k, does the system have no solution or infinitely many? (c) For k = 5, solve with an inverse.
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