Linear Systems and Matrices: math test, College

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Math tests College : Linear Systems and Matrices — Zyro the alien explorer of Planète Maths

Suggested time: 45 minutes. Out of 20 points. Calculator allowed only when the problem says so.

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1 Gaussian elimination / 4 pts

Solve by Gaussian elimination, showing each row operation: \(2x + y - z = 1,\ x - y + 2z = 5,\ 3x + 2y + z = 10\).

2 Free variable / 3 pts

Solve \(x + y + 2z = 3,\ 2x + 2y + 5z = 8,\ x + y + z = 1\). Take \(y\) as the free variable.

3 Matrix operations / 3 pts

Let \(A = \begin{pmatrix} 1 & -1 \\ 2 & 3 \end{pmatrix}\) and \(B = \begin{pmatrix} 0 & 2 \\ -1 & 4 \end{pmatrix}\). Compute \(2A - B\), \(AB\), and \(BA\).

4 Inverse and solution / 4 pts

Let \(A = \begin{pmatrix} 3 & 2 \\ 4 & 3 \end{pmatrix}\). (a) Compute \(\det A\) and decide whether \(A\) is invertible. (b) Find \(A^{-1}\). (c) Solve \(3x + 2y = 7,\ 4x + 3y = 10\).

5 A 3 by 3 determinant / 3 pts

Let \(A = \begin{pmatrix} 1 & 0 & 2 \\ 3 & 1 & -1 \\ 2 & 4 & 0 \end{pmatrix}\). Compute \(\det A\), then \(\det(A^T)\) and \(\det(2A)\). Is \(A\) invertible?