Matrices and determinants – university Year 1 maths worksheet with answers to download as PDF

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

1. Compute \(AB\) and \(BA\). [4 marks]
2. What does multiplying by \(B\) on the right and on the left do to \(A\)? [2 marks]
3. Compute \(A^2\). [2 marks]
4. Is \((A+B)^2=A^2+2AB+B^2\) in this case? Explain. [3 marks]

Let \(C=\) \(\begin{pmatrix}3&1\\5&2\end{pmatrix}\) and \(D=\) \(\begin{pmatrix}2&4\\1&2\end{pmatrix}\).

1. Find \(\det C\) and \(C^{-1}\). [3 marks]
2. Find \(\det D\). What does it tell you about \(D\)? [2 marks]
3. Find all values of \(k\) for which \(\begin{pmatrix}1&k\\k&4\end{pmatrix}\) has no inverse. [3 marks]

Evaluate each determinant, using the method indicated.

1. \(\begin{pmatrix}2&0&1\\1&3&2\\1&1&2\end{pmatrix}\), expanding along the first row. [4 marks]
2. \(\begin{pmatrix}2&1&7\\0&3&5\\0&0&-4\end{pmatrix}\) [2 marks]
3. \(\begin{pmatrix}1&2&3\\4&5&6\\7&8&9\end{pmatrix}\) using row operations. [4 marks]

Solve each system by the method given.

1. \(2x+y=5\) and \(5x+3y=13\), using the inverse of the coefficient matrix. [4 marks]
2. \(2x+y=5\) and \(x+3y=10\), using Cramer's rule. [4 marks]

Let \(A\) be a \(3\times3\) matrix with \(\det A=5\) and \(B\) a \(3\times3\) matrix with \(\det B=-2\). Find each value, naming the property you use.

1. \(\det(AB)\) [2 marks]
2. \(\det(2A)\) [2 marks]
3. \(\det(A^T)\) and \(\det(A^{-1})\) [3 marks]
4. \(\det(A^2B^{-1})\) [3 marks]

Let \(A=\) \(\begin{pmatrix}1&2\\3&4\end{pmatrix}\) and \(B=\) \(\begin{pmatrix}0&1\\1&0\end{pmatrix}\) as before.

1. Verify that \((AB)^T=B^TA^T\). [3 marks]
2. Show that \((AB)^{-1}=B^{-1}A^{-1}\) for these matrices. [3 marks]
3. The matrix \(R=\) \(\begin{pmatrix}0&-1\\1&0\end{pmatrix}\) is a rotation through \(90^\circ\). Find \(\det R\) and \(R^2\). [3 marks]
4. Find the area of the triangle with vertices \((0,0)\), \((4,1)\) and \((2,5)\). [3 marks]

At a student café, 3 coffees and 2 cakes cost £11.70, and 2 coffees and 3 cakes cost £11.05. Let a coffee cost £\(c\) and a cake cost £\(k\).

1. Write the information as a matrix equation \(M\mathbf x=\mathbf b\).
2. Find \(\det M\) and \(M^{-1}\).
3. Solve the system.
4. How much do 5 coffees and 4 cakes cost?

A small shop sells three products: milk, bread and eggs. The numbers sold on three days and the unit prices (in £) are in the tables.

1. Write the sales as a \(3\times3\) matrix \(S\) and the prices as a column vector \(\mathbf p\).
2. Compute \(S\mathbf p\) and interpret each entry.
3. Find the total takings over the three days.
4. Prices rise by 10 per cent. Write the new takings as a matrix product.

Each year, 5% of a city's residents move to the suburbs and 3% of suburban residents move to the city. Let \(\mathbf v_n=\) \(\begin{pmatrix}c_n\\s_n\end{pmatrix}\) give the populations (in thousands) after \(n\) years, so \(\mathbf v_{n+1}=P\mathbf v_n\) with \(P=\) \(\begin{pmatrix}0.95&0.03\\0.05&0.97\end{pmatrix}\).

1. Check that each column of \(P\) sums to 1 and explain why.
2. Initially the city has 600 thousand and the suburbs 400 thousand. Find the populations after one year.
3. Find the populations after 10 years (use a calculator for \(P^{10}\)).
4. Find the long-term populations by solving \(P\mathbf v=\mathbf v\) with \(c+s=1000\).

In a video game, a triangular shield has vertices \((0,0)\), \((2,0)\) and \((0,3)\). The game applies the matrix \(T=\) \(\begin{pmatrix}0&-2\\2&0\end{pmatrix}\) to every point.

1. Find the images of the three vertices.
2. Find \(\det T\) and use it to find the area of the new shield.
3. Describe \(T\) geometrically as a rotation followed by an enlargement.
4. What matrix would undo the transformation?

Agents encode pairs of letters (\(A=0,B=1,\dots,Z=25\)) with the matrix \(K=\) \(\begin{pmatrix}3&3\\2&5\end{pmatrix}\), working modulo 26. The pair \((7,8)\) stands for "HI".

1. Show that \(\det K=9\) and that \(9\times3\equiv1\pmod{26}\).
2. Encode "HI".
3. Show that the decoding matrix is \(K^{-1}\equiv\) \(\begin{pmatrix}15&17\\20&9\end{pmatrix}\) (mod 26).
4. Decode the pair \((19,2)\) to check.

A cook mixes oats, nuts and raisins to make 100 g of trail mix. Let the masses be \(x\), \(y\) and \(z\) grams. Her targets are: total 100 g; "energy points" \(2x+5y+z=270\); and \(x+3z=110\).

1. Write the system as \(M\mathbf x=\mathbf b\).
2. Show that \(\det M=5\), so the solution is unique.
3. Solve by row reduction.
4. Check that your answer satisfies all three conditions.

Kirchhoff's laws for a circuit give the equations \(I_1-I_2-I_3=0\), \(2I_1+4I_2=14\) and \(4I_2-6I_3=2\), where the currents are in amperes.

1. Write the equations as a matrix equation.
2. Evaluate the determinant of the coefficient matrix.
3. Solve for the three currents.
4. Check that the current law holds at the junction.

An economy has two industries, steel and energy. To produce 1 unit of steel, 0.2 units of steel and 0.4 units of energy are used up. To produce 1 unit of energy, 0.3 units of steel and 0.1 units of energy are used up. The consumption matrix is \(A=\) \(\begin{pmatrix}0.2&0.3\\0.4&0.1\end{pmatrix}\). The outside demand is \(\mathbf d=(100,50)\). The production vector \(\mathbf x\) must satisfy \(\mathbf x=A\mathbf x+\mathbf d\).

1. Show that \((I-A)\mathbf x=\mathbf d\).
2. Find \(I-A\) and its determinant.
3. Find \((I-A)^{-1}\) and hence \(\mathbf x\).
4. Interpret your answer in context: why is the production larger than the demand?

Fibonacci numbers satisfy \(F_{n+1}=F_n+F_{n-1}\) with \(F_0=0\), \(F_1=1\). In a rabbit model, \(F_n\) is the number of pairs after \(n\) months. Let \(Q=\) \(\begin{pmatrix}1&1\\1&0\end{pmatrix}\).

1. Compute \(Q^2\) and \(Q^3\) and compare their entries with the table.
2. Conjecture a formula for \(Q^n\) and prove it by induction.
3. Find \(Q^{10}\) and read off \(F_{10}\).
4. Find \(\det Q^n\) and deduce a relation among \(F_{n-1}\), \(F_n\) and \(F_{n+1}\) (Cassini's identity).

A weather model has states Sunny, Cloudy and Rainy. Each column of the transition matrix \(W\) gives the probabilities for tomorrow, given today's state: \(W=\) \(\begin{pmatrix}0.7&0.3&0.1\\0.2&0.5&0.4\\0.1&0.2&0.5\end{pmatrix}\).

1. Check that \(W\) is a valid transition matrix.
2. Today is sunny, so \(\mathbf v_0=(1,0,0)^T\). Find \(\mathbf v_1\) and \(\mathbf v_2\).
3. Find the long-term proportions by solving \(W\mathbf v=\mathbf v\) with \(v_1+v_2+v_3=1\).
4. On how many days out of 100 will it rain in the long run?

A graphics engine applies two transformations to every point: first the reflection \(S=\) \(\begin{pmatrix}1&0\\0&-1\end{pmatrix}\) in the \(x\)-axis, then the rotation \(R=\) \(\begin{pmatrix}0&-1\\1&0\end{pmatrix}\) through \(90^\circ\) anticlockwise.

1. Find the matrix \(RS\) of the combined transformation (first \(S\) then \(R\)).
2. Find the matrix \(SR\) (first \(R\) then \(S\)) and show that it is different.
3. Find the determinants of \(S\), \(R\) and \(RS\). What does a negative determinant mean?
4. Find the image of the point \((3,1)\) under each combined transformation.

Two sensors give readings related to an unknown pair \((x,y)\) by \(x+ky=2\) and \(kx+4y=4\), where \(k\) is a calibration constant.

1. Show that the determinant of the coefficient matrix is \(4-k^2\).
2. For which \(k\) is there a unique solution? Find it in terms of \(k\).
3. Find what happens when \(k=2\).
4. Find what happens when \(k=-2\). Compare with part (c).

The edges of a crystal unit cell are given by the vectors \(\mathbf a=(2,0,1)\), \(\mathbf b=(1,3,2)\) and \(\mathbf c=(1,1,2)\) (in ångströms).

1. Write the \(3\times3\) matrix whose rows are \(\mathbf a\), \(\mathbf b\), \(\mathbf c\).
2. Find its determinant, and hence the volume of the cell.
3. What is the volume of the tetrahedron with these three edges from one vertex?
4. If the cell is stretched by a factor of 1.5 in every direction, what is the new volume?

For a \(2\times2\) matrix \(A\) the Cayley-Hamilton theorem states \(A^2-(\operatorname{tr}A)A+(\det A)I=0\). Let \(A=\) \(\begin{pmatrix}1&2\\3&4\end{pmatrix}\).

1. Find \(\operatorname{tr}A\) and \(\det A\), and verify the theorem.
2. Use it to express \(A^{-1}\) as a combination of \(A\) and \(I\).
3. Express \(A^3\) in the form \(pA+qI\).
4. Check part (c) by direct multiplication.