How to add and subtract matrices and solve matrix equations

Adding or subtracting matrices means combining entries in matching positions. The matrices need the same dimensions so every entry has a partner. We can use those same operations to solve an equation for an unknown matrix.

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Match the dimensions and positions

$A=\left[\begin{array}{rr}1& 2\\ 3& 4\end{array}\right]$

$B=\left[\begin{array}{rr}5& 6\\ 7& 8\end{array}\right]$

Both matrices are $2\times 2$. Add each entry of $A$ to the entry in the same row and column of $B$.

$A+B=\left[\begin{array}{rr}6& 8\\ 10& 12\end{array}\right]$

Subtraction uses the same positions, in the stated order.

$A-B=\left[\begin{array}{rr}-4& -4\\ -4& -4\end{array}\right]$

Isolate an unknown matrix

If $X+A=A+B$, subtract $A$ from both sides to get $X=B$. Entry by entry, the matrix with rows $(6,8)$ and $(10,12)$ minus $A$ gives rows $(5,6)$ and $(7,8)$.

Solve a matrix equation with several terms

Let the equation be $P+Q=X+C-D$, using these four known matrices.

$P=\left[\begin{array}{rr}5& -7\\ -1& 0\end{array}\right]$

$Q=\left[\begin{array}{rr}3& -4\\ -1& -6\end{array}\right]$

$C=\left[\begin{array}{rr}1& 0\\ 17& 0\end{array}\right]$

$D=\left[\begin{array}{rr}2& 4\\ 11& -9\end{array}\right]$

The sum $P+Q$ has rows $(8,-11)$ and $(-2,-6)$. The difference $C-D$ has rows $(-1,-4)$ and $(6,9)$. Subtract that difference from both sides to isolate $X$.

$X=\left[\begin{array}{rr}9& -7\\ -8& -15\end{array}\right]$

Use the addition properties

Matrix addition is commutative and associative: $A+B=B+A$ and $(A+B)+C=A+(B+C)$ when the dimensions match. Subtraction generally changes when its order or grouping changes, so we keep the signs and parentheses from the original expression.

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