Question 9
Consider the matrix equation.
$$\mathbf{X}\begin{bmatrix}0&0&1\\0&1&1\\1&1&1\end{bmatrix}=\begin{bmatrix}1&2&2\\2&1&2\\2&2&1\end{bmatrix}$$
Matrix $\mathbf{X}$ is
That’s it. Step through the hints below to see why. In 2022, 72.2% of Queensland students chose (A).
Not quite. Let $\mathbf{A}$ be the matrix beside $\mathbf{X}$ and $\mathbf{B}$ the right-hand side. This option is $\mathbf{A}^{-1}\mathbf{B}$, with the inverse on the wrong side. Because $\mathbf{X}$ sits on the left of $\mathbf{A}$, the inverse goes on the right: $\mathbf{X}=\mathbf{B}\mathbf{A}^{-1}$. In 2022, 20.3% of Queensland students chose (B).
Not quite. Let $\mathbf{A}$ be the matrix beside $\mathbf{X}$ and $\mathbf{B}$ the right-hand side. This option is $\mathbf{A}\mathbf{B}$, the two given matrices multiplied together. To undo $\mathbf{A}$ you multiply by its inverse: $\mathbf{X}=\mathbf{B}\mathbf{A}^{-1}$. In 2022, 3.3% of Queensland students chose (C).
Not quite. Let $\mathbf{A}$ be the matrix beside $\mathbf{X}$ and $\mathbf{B}$ the right-hand side. This option is $\mathbf{B}\mathbf{A}$. The order is right, but multiplying by $\mathbf{A}$ a second time doesn’t undo it. Multiply by $\mathbf{A}^{-1}$ instead: $\mathbf{X}=\mathbf{B}\mathbf{A}^{-1}$. In 2022, 3.5% of Queensland students chose (D).
Let $\mathbf{A}$ be the matrix beside $\mathbf{X}$ and $\mathbf{B}$ the right-hand side, so the equation is $\mathbf{X}\mathbf{A}=\mathbf{B}$. Put an option in for $\mathbf{X}$ and multiply. Entries that differ from $\mathbf{B}$ show in bold blue.
Every entry matches $\mathbf{B}$, so this option satisfies the equation in the question.
Every entry matches here, but this checks $\mathbf{A}\mathbf{X}=\mathbf{B}$. The question has $\mathbf{X}$ on the left of $\mathbf{A}$, so this option solves a different equation.
Some entries differ from $\mathbf{B}$, so this option doesn’t satisfy the equation with this order of multiplication.
Let $\mathbf{A}$ be the matrix beside $\mathbf{X}$ and $\mathbf{B}$ the right-hand side. Multiply both sides by $\mathbf{A}^{-1}$ on the right, the side $\mathbf{A}$ is on. Matrix multiplication isn’t commutative, so the side matters.
$$\begin{aligned}\mathbf{X}\mathbf{A}\mathbf{A}^{-1}&=\mathbf{B}\mathbf{A}^{-1}\\\mathbf{X}&=\mathbf{B}\mathbf{A}^{-1}\end{aligned}$$
Find $\mathbf{A}^{-1}$ on your calculator. It exists because $\det\mathbf{A}=-1\ne0$.
$$\mathbf{A}^{-1}=\begin{bmatrix}0&-1&1\\-1&1&0\\1&0&0\end{bmatrix}$$
Multiply with $\mathbf{B}$ first. On the calculator, enter $\mathbf{B}\times\mathbf{A}^{-1}$.
$$\mathbf{X}=\mathbf{B}\mathbf{A}^{-1}$$
$$\begin{bmatrix}1&2&2\\2&1&2\\2&2&1\end{bmatrix}\begin{bmatrix}0&-1&1\\-1&1&0\\1&0&0\end{bmatrix}$$
$$=\begin{bmatrix}0&1&1\\1&-1&2\\-1&0&2\end{bmatrix}$$
With $\mathbf{A}$ the matrix beside $\mathbf{X}$ and $\mathbf{B}$ the right-hand side, multiply both sides by $\mathbf{A}^{-1}$ on the right.
$$\mathbf{X}=\mathbf{B}\mathbf{A}^{-1}$$
$$=\begin{bmatrix}1&2&2\\2&1&2\\2&2&1\end{bmatrix}\begin{bmatrix}0&-1&1\\-1&1&0\\1&0&0\end{bmatrix}$$
$$=\begin{bmatrix}0&1&1\\1&-1&2\\-1&0&2\end{bmatrix}$$
Check: multiplying this matrix by $\mathbf{A}$ gives back $\mathbf{B}$.
How Queensland students answered in 2022
From the QCAA Specialist Mathematics subject report for 2022. The percentages can add to less than 100 because some students didn’t answer.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording is from the 2022 QCAA Specialist Mathematics external assessment, © State of Queensland (QCAA) 2022, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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