Matrices and Systems of Equations: math test, Grade 12

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Math tests Grade 12 : Matrices and Systems of Equations — Zyro the alien explorer of Planète Maths

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

Sommaire

1 Operations / 3 pts

Let \(P = \begin{bmatrix}4&-1\\2&0\end{bmatrix}\) and \(Q = \begin{bmatrix}3&5\\-2&1\end{bmatrix}\). Find \(2P - Q\) and \(PQ\).

2 Determinants / 3 pts

Find (a) \(\det \begin{bmatrix}7&3\\2&1\end{bmatrix}\) and (b) \(\det \begin{bmatrix}1&0&2\\3&1&-1\\0&2&4\end{bmatrix}\).

3 Inverse and system / 3 pts

Let \(M = \begin{bmatrix}5&2\\7&3\end{bmatrix}\). (a) Find \(M^{-1}\). (b) Solve \(M\begin{bmatrix}x\\y\end{bmatrix} = \begin{bmatrix}4\\9\end{bmatrix}\).

4 Gaussian elimination / 4 pts

Solve by Gaussian elimination: \(x + 2y + z = 4\), \(2x - y + 3z = -3\), \(3x + y - z = 6\).

5 Cramer’s Rule / 3 pts

Use Cramer’s Rule to solve \(3x + 2y = 3\) and \(5x - y = -8\).

6 A transformation of a square / 4 pts

The square \(OABC\) has vertices \(O(0, 0)\), \(A(2, 0)\), \(B(2, 2)\), \(C(0, 2)\). Apply the shear \(S = \begin{bmatrix}1&2\\0&1\end{bmatrix}\). (a) Find the images of the vertices. (b) Find the area of the image without drawing it, and explain.