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Solve a matrix equation

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Calculate the determinant of the matrix A shown below

\[A=\begin{bmatrix}2&1\\1&1\end{bmatrix},\quad \det(A)=2\times1-1\times1=1\]
Year 12, VCE Units 3 & 4, Chapter 10, 4 difficulty levels, Calculator allowedBuild a worksheet

Solving a matrix equation

First remove any additive matrix, then identify the side on which the coefficient multiplies the unknown. If the coefficient matrix is invertible, multiply by its inverse on that same side and preserve the order of every remaining factor

\[AX=B\Rightarrow X=A^{-1}B,\qquad XA=B\Rightarrow X=BA^{-1}\]
  1. Remove additive terms Subtract an additive matrix from both sides before applying an inverse \(\displaystyle AX+C=B\quad\Rightarrow\quad AX=B-C\)
  2. Read the coefficient side Identify whether the coefficient multiplies the unknown on the left or on the right \(\displaystyle AX=B\quad\text{or}\quad XA=B\)
  3. Multiply on the same side Use the inverse on the side that cancels the coefficient matrix \(\displaystyle AX=B\quad\Rightarrow\quad A^{-1}AX=A^{-1}B\quad\Rightarrow\quad X=A^{-1}B\)
  4. Keep product order Matrix products cannot generally be rearranged because matrix multiplication is not commutative \(\displaystyle A^{-1}B\ne BA^{-1}\text{ in general}\)

Worked example

Solve the matrix equation below

\[AX=B,\quad A=\begin{bmatrix}3&1\\1&1\end{bmatrix},\quad B=\begin{bmatrix}5&15\\1&7\end{bmatrix}\]
  1. \(\displaystyle AX=B\)The coefficient matrix is on the left of the unknown, so the inverse will also be used on the left
  2. \(\displaystyle \det(A)=3\times1-1\times1=2,\quad A^{-1}=\frac{1}{2}\begin{bmatrix}1&-1\\-1&3\end{bmatrix}\)The non-zero determinant allows the inverse to be formed with the 2 × 2 inverse rule
  3. \(\displaystyle A^{-1}AX=A^{-1}B\quad\Rightarrow\quad X=A^{-1}B\)Multiplying on the left makes A⁻¹A equal the identity matrix
  4. \(\displaystyle X=\frac{1}{2}\begin{bmatrix}1&-1\\-1&3\end{bmatrix}\begin{bmatrix}5&15\\1&7\end{bmatrix}=\frac{1}{2}\begin{bmatrix}4&8\\-2&6\end{bmatrix}=\begin{bmatrix}2&4\\-1&3\end{bmatrix}\)Multiplying the inverse by B gives the unknown matrix

Common mistakes

Multiplying the inverse on the wrong side
✕\(\displaystyle AX=B\quad\Rightarrow\quad X=BA^{-1}\)✓\(\displaystyle AX=B\quad\Rightarrow\quad X=A^{-1}B\)
The inverse must multiply on the same side as the coefficient so that the coefficient and its inverse cancel
Assuming matrix multiplication is commutative
✕\(\displaystyle A^{-1}B=BA^{-1}\)✓\(\displaystyle A^{-1}B\ne BA^{-1}\text{ in general}\)
Changing the order of matrix factors usually changes the product, so the order needed to cancel A must be preserved
Reversing an additive subtraction
✕\(\displaystyle AX+C=B\quad\Rightarrow\quad X=A^{-1}(C-B)\)✓\(\displaystyle AX+C=B\quad\Rightarrow\quad X=A^{-1}(B-C)\)
Subtract C from both sides in the written order before multiplying by the inverse

Difficulty levels

introIsolate an unknown from an additive matrix equation
standardSolve a left-coefficient equation with an integer inverse effect
challengeSolve a right-coefficient or additive equation with exact rational entries
examChoose the only isolation formula that preserves the required factor order

Questions about this topic

Which side should the inverse multiply on?+

Use the same side on which the coefficient multiplies the unknown: the left for a left coefficient, or the right for a right coefficient

What should be done first when a matrix is added to the product?+

Subtract the added matrix from both sides, then multiply on the left by the inverse of the coefficient matrix

Why cannot matrix factors be rearranged?+

Matrix multiplication is not generally commutative, so changing the order can change the product or make the product undefined

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 12, VCE Units 3 & 4, General Mathematics, chapter 10: Matrices

In this chapter

State the order of a matrix and locate its elementsIdentify row, column, square, symmetric and zero matricesRead information from a labelled data matrixRead, transpose and classify a matrixConstruct a matrix from labelled information or an entry rule10A Matrix notation, types and construction: Solve a matrix equation10A Matrix notation, types and construction: Construct, apply and invert a permutation matrixAdd and subtract matricesFind unknown entries in a matrix equationMultiply a matrix by a scalarCombine scalar multiples of matrices10B Linear operations on matrices: Determine whether an inverse exists and calculate itDecide whether a matrix product is defined and give its orderMultiply matricesEvaluate powers of a square matrixIdentify and construct the identity matrix10C Matrix products, powers and permutations: Construct, apply and invert a permutation matrixFind the determinant of a 2 × 2 matrixFind the inverse of a 2 × 2 matrixWrite simultaneous equations as a matrix equationSolve simultaneous equations using the inverse matrix10D Determinants, inverses and matrix equations: Determine whether an inverse exists and calculate it10D Determinants, inverses and matrix equations: Solve a matrix equationUse matrix multiplication to answer an application questionUse row and column matrices of ones to extract totalsRepresent a network in a matrixIdentify the network a connection matrix describesInterpret connection sums and symmetryCount routes using powers of a connection matrixConstruct and analyse relation matrices