This test has 8 questions worth 20 points in total and should take about 45 minutes. It covers matrix equations, 2 by 2 inverses, solving systems with matrices and transformations of the plane. Show all work. A calculator is allowed.
- Grade: Precalculus
- Chapter: Solving Systems with Matrices
- Common Core: A-REI.C.8, A-REI.C.9, N-VM.C.11, N-VM.C.12
- Time: 45 minutes
- Total: 20 points (Question 1: 2 pts; Question 2: 3 pts; Question 3: 3 pts; Question 4: 2 pts; Question 5: 2 pts; Question 6: 3 pts; Question 7: 2 pts; Question 8: 3 pts)
Question 1: Matrix form of a system (2 points)
Consider the system \(5x-2y=9\), \(x+3y=-5\).
- Write the system as a matrix equation \(AX=B\). (1 point)
- Find \(\det A\) and state whether \(A\) has an inverse. (1 point)
Question 2: Inverse of a matrix (3 points)
Matrix \(A\) has rows \(4,\ 5\) and \(3,\ 4\).
- Find \(\det A\). (1 point)
- Find \(A^{-1}\). (1 point)
- Show that \(AA^{-1}\) is the identity matrix. (1 point)
Question 3: Solve a system with matrices (3 points)
Consider the system \(3x+2y=4\), \(7x+5y=9\).
- Find the inverse of the coefficient matrix. (1 point)
- Solve the system using matrices. (2 points)
Question 4: Singular matrices (2 points)
Matrix \(C\) has rows \(2,\ k\) and \(3,\ 6\).
- Find the value of \(k\) for which \(C\) has no inverse. (1 point)
- For that \(k\), does \(2x+ky=6\), \(3x+6y=9\) have no solution, one solution or infinitely many? Explain. (1 point)
Question 5: Transforming points (2 points)
Matrix \(Q\) has rows \(0,\ 1\) and \(-1,\ 0\). The points \((3,\ 1)\) and \((-2,\ 4)\) are plotted.
- Find the images of the two points under \(Q\). (1 point)
- Describe the transformation geometrically. (1 point)
Question 6: Image of the unit square (3 points)
The unit square is shown. Matrix \(N\) has rows \(3,\ 1\) and \(0,\ 2\).
- Find the images of the four vertices under \(N\). (2 points)
- Find the area of the image. (1 point)
Question 7: Area under a matrix (2 points)
The triangle shown has area \(7\). It is transformed by the matrix \(T\) with rows \(2,\ -1\) and \(3,\ 4\).
- Find \(\det T\). (1 point)
- Find the area of the image triangle. (1 point)
Question 8: Ticket prices (3 points)
At the school play, Noah buys 2 adult tickets and 3 student tickets for \(\$36\). Sofia buys 3 adult tickets and 2 student tickets for \(\$39\).
- Write a matrix equation for this situation. (1 point)
- Find the price of each type of ticket using matrices. (2 points)
See every chapter on the Precalculus math page.