Solve a matrix equation

Solving a matrix equation

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

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 shown below

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

Difficulty levels

Questions about this topic

In this chapter