Question 16
Let $\mathbf{A}=\begin{bmatrix}1&2&2\\2&1&1\\2&2&1\end{bmatrix}$
Given $\mathbf{A}^4=p\mathbf{A}^3+q\mathbf{A}^2+r\mathbf{A}+3\mathbf{I}$, use matrix algebra to determine the value of the scalars $p$, $q$ and $r$.
[6 marks]One matrix equation hides nine ordinary equations
Watch the highlighted boxes. The same position in every matrix gives one equation in $p$, $q$ and $r$. Three positions give three equations.
Two matrices are equal only when every pair of matching entries is equal. Row 1 is one convenient set of three entries, and $3\mathbf{I}$ adds 3 only on the main diagonal.
Choose your own three entries
The question doesn’t say which entries to compare. Pick any three entries of $\mathbf{A}^4$ and the same positions light up in every matrix. Each position gives one equation. Some choices pin down $p$, $q$ and $r$ and some don’t, and the determinant tells you which.
Pick three entries. Each one gives you one equation, and three unknowns need three equations.
At least two of your entries give exactly the same equation. That leaves fewer than three different equations for three unknowns, so the determinant is 0 and there is no unique solution. Swap one of the matching entries.
These three equations are all different, but the determinant is still 0. One of them is a combination of the other two, so it tells you nothing new.
The determinant is not 0, so the coefficient matrix has an inverse and there is exactly one solution. Every choice that works gives these same values, because $p=2$, $q=10$ and $r=10$ make all nine entries match.
This is row 1, the choice the marking guide uses.
Work through the solution one mark at a time
Try each step yourself before you reveal it. The cards follow the marking guide, which compares the entries in row 1.
Enter $\mathbf{A}$ into your calculator and find its powers.
Substitute them into the given equation, with $3\mathbf{I}$ written out in full.
$$\begin{aligned}&\begin{bmatrix}177&176&144\\144&145&120\\176&176&145\end{bmatrix}\\&\quad=p\begin{bmatrix}37&38&32\\32&31&25\\38&38&31\end{bmatrix}\\&\qquad+q\begin{bmatrix}9&8&6\\6&7&6\\8&8&7\end{bmatrix}\\&\qquad+r\begin{bmatrix}1&2&2\\2&1&1\\2&2&1\end{bmatrix}\\&\qquad+\begin{bmatrix}3&0&0\\0&3&0\\0&0&3\end{bmatrix}\end{aligned}$$
Any equivalent way of writing this equation is accepted. Calculating the matrices for $\mathbf{A}^4$, $\mathbf{A}^3$ and $\mathbf{A}^2$ is also accepted for this mark, and it can be implied by your later working.
Marker: correctly substitutes results into given equation to form an equation based on $3\times3$ matrices
Two matrices are equal only when their matching entries are equal. Equate the three entries in row 1.
$$\begin{aligned}177&=37p+9q+r+3\\176&=38p+8q+2r+0\\144&=32p+6q+2r+0\end{aligned}$$
Move each constant to the right-hand side.
$$\begin{aligned}37p+9q+r&=174\\38p+8q+2r&=176\\32p+6q+2r&=144\end{aligned}$$
Any equivalent form of the three equations is accepted. This mark can be implied by your later working, and if a power of $\mathbf{A}$ was wrong you can still earn it by follow-through. Row 1 is not the only choice, but the three entries must give three independent equations.
Marker: uses equation based on $3\times3$ matrices to establish 3 equations in 3 unknowns
Write the system as one matrix equation: the matrix of coefficients, the column of unknowns and the column of constants.
$$\begin{bmatrix}37&9&1\\38&8&2\\32&6&2\end{bmatrix}\begin{bmatrix}p\\q\\r\end{bmatrix}=\begin{bmatrix}174\\176\\144\end{bmatrix}$$
Any equivalent equation in matrix form is accepted.
Marker: expresses 3 equations in matrix form
The coefficient matrix has determinant 12, so its inverse exists. Multiply both sides on the left by the inverse.
$$\begin{bmatrix}p\\q\\r\end{bmatrix}=\begin{bmatrix}37&9&1\\38&8&2\\32&6&2\end{bmatrix}^{-1}\begin{bmatrix}174\\176\\144\end{bmatrix}$$
The question asks for matrix algebra, so write this line even though the calculator does the arithmetic. Any equivalent method that shows matrix algebra is accepted, and this mark can be implied by your later working.
Marker: demonstrates the use of matrix algebra in solving the matrix equation
Evaluate the product with your calculator.
$$\begin{bmatrix}p\\q\\r\end{bmatrix}=\begin{bmatrix}2\\10\\10\end{bmatrix}$$
Any equivalent way of stating the values is accepted, such as $p=2$, $q=10$, $r=10$. To check, enter $2\mathbf{A}^3+10\mathbf{A}^2+10\mathbf{A}+3\mathbf{I}$ and compare it with $\mathbf{A}^4$.
Marker: determines values of $p$, $q$ and $r$
The last mark is for how you set out your working. The marker looks for suitable matrix notation, a logical order and correct use of equals signs, at least as far as the start of the matrix equation in Step 3. Writing the substituted matrix equation, the three equations and the matrix equation on separate lines covers this.
Marker: shows logical organisation, communicating key steps to at least the start of expressing the 3 equations in matrix form
Each phrase of the question gave you something
The question doesn’t tell you to calculate the powers of $\mathbf{A}$ or which entries to compare. You have to see that one matrix equation holds entry by entry and choose three entries that give independent equations. Those equations then go into a second procedure, solving a system with an inverse matrix. It is not complex unfamiliar, because “use matrix algebra” points straight at the inverse method you have practised for systems of equations.
If I had to guess where you might lose marks, it would be the matrix algebra itself. The subject report for this exam recommends “supporting students to make connections between the subject matter and the glossary terms so that terminology such as use ‘matrix algebra’ to solve a system of equations is familiar to students”. In the marking guide, one mark is for writing the matrix equation and another is for showing matrix algebra as you solve it (the inverse step in the sample response), and the communication mark needs your working to reach the matrix form. A list of three equations followed by values from an equation solver shows none of that. The other risk is a set of entries that repeat or depend on each other, so check the determinant before you trust the inverse.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording and marking-guide steps are from the 2021 QCAA Specialist Mathematics external assessment, © State of Queensland (QCAA) 2021, licensed under CC BY 4.0, and have been adapted. Tangent Tuition is not affiliated with the QCAA.
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