Worked examples — Matrix operations — addition, subtraction (conditions)

This page is the drill hall for matrix addition and subtraction. The parent note built the rule: two matrices combine element-by-element, and only if they have the same order (same number of rows and same number of columns). Here we throw every kind of situation at that rule so you never meet a surprise in an exam.

Before we start, one reminder built from zero: a matrix like $A=[a_{ij}{]}_{m\times n}$ is just a rectangle of numbers. The little label $a_{ij}$ means "the number sitting in row $i$, column $j$". So $a_{23}$ is the number in the 2nd row, 3rd column. When we "add corresponding entries", we mean: the number in slot (2,3) of $A$ plus the number in slot (2,3) of $B$ lands in slot (2,3) of the answer. See Matrix Notation and Terminology if that indexing feels shaky.

The scenario matrix

Every problem this topic can throw at you falls into one of these cells. The rest of the page works an example for each.

C1 · Mixed signs, valid order

Forecast: cover the answer and guess the top-left entry. (It's $3+(-1)$.)

1. Check order. Why this step? Addition is only defined for identical orders — the very first thing to test, always. Both are $2\times 2$. ✓ Defined.
2. Add slot by slot. Why this step? By definition $c_{ij}=a_{ij}+b_{ij}$, so each answer entry only ever looks at the same slot in each matrix. $A+B=\left[\begin{array}{cc}3+(-1)& -5+4\\ -2+6& 0+(-7)\end{array}\right]=\left[\begin{array}{cc}2& -1\\ 4& -7\end{array}\right]$
3. Watch the sign rules. Why this step? Adding a negative is subtracting; $-5+4=-1$, $0+(-7)=-7$. The most common slip is here, not in the layout.

Verify: Add in the other order $B+A$: $-1+3=2$, $4-5=-1$, $6-2=4$, $-7+0=-7$ — identical, as commutativity promises.

C2 · Invalid order — the answer is a word

Forecast: both have three numbers — tempting, right?

1. Read the orders. Why this step? Never count elements; count shape. $M$ is $1\times 3$ (one row, three columns). $N$ is $3\times 1$ (three rows, one column).
2. Compare. Why this step? Addition needs $m$ and $n$ to match. Here $1\times 3\ne 3\times 1$. ✗
3. State the result. $M+N$ is undefined. There is no slot $(1,3)$ in $N$ (it has only one column), and no slot $(3,1)$ in $M$ (it has only one row).

Verify: Total element count agrees ($3=3$) yet the operation still fails — proof that "same count" is a trap. This is exactly Mistake 1 from the parent note.

C3 · The zero matrix (additive identity)

Forecast: guess before reading.

1. Order check. Why this step? $O$ must be the same order $2\times 2$ to be the identity here — the zero matrix comes in every size (see Zero Matrix and Identity Matrix).
2. Add. Why this step? $a_{ij}+0=a_{ij}$ for every real number, so nothing changes. $A+O=\left[\begin{array}{cc}7+0& -4+0\\ 2+0& 9+0\end{array}\right]=\left[\begin{array}{cc}7& -4\\ 2& 9\end{array}\right]=A$

Verify: Result equals $A$ entry-for-entry. $O$ is the additive identity, mirroring how $0$ behaves for ordinary numbers.

C4 · The additive inverse

Forecast: what should the sum be?

1. Negate every entry. Why this step? $-A=[-a_{ij}]$ — flip the sign of each slot (this is scalar multiplication by $-1$, see Scalar Multiplication of Matrices). $-A=\left[\begin{array}{cc}-5& 3\\ 0& -8\end{array}\right]$
2. Add. Why this step? $a_{ij}+(-a_{ij})=0$ for every slot, so we should land on the zero matrix. $A+(-A)=\left[\begin{array}{cc}5-5& -3+3\\ 0+0& 8-8\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]=O$

Verify: Every entry is $0$. This is why $A-B=A+(-B)$: subtraction is just adding the inverse.

C5 · Degenerate shapes (thin matrices)

Forecast: does the rule change for skinny matrices? (It doesn't.)

1. (a) Two $1\times 3$ rows. Why this step? Same order → element-wise. $\left[\begin{array}{ccc}2+3& 5-5& -1+4\end{array}\right]=\left[\begin{array}{ccc}5& 0& 3\end{array}\right]$.
2. (b) Two $2\times 1$ columns. Why this step? Subtract each slot. $\left[\begin{array}{c}4-1\\ -2-(-6)\end{array}\right]=\left[\begin{array}{c}3\\ 4\end{array}\right]$.
3. (c) Two $1\times 1$ matrices. Why this step? A $1\times 1$ matrix is one number wearing a box — addition collapses to ordinary arithmetic: $9+(-4)=5$, i.e. $\left[\begin{array}{c}5\end{array}\right]$.

Verify: (a) middle entry $5-5=0$; (b) bottom $-2+6=4$; (c) $9-4=5$. All consistent — the rule scales down cleanly to vectors and scalars.

C6 · Order sensitivity of subtraction

Forecast: are they equal? Opposite? Unrelated?

1. Compute $A-B$. Why this step? $d_{ij}=a_{ij}-b_{ij}$. $A-B=\left[\begin{array}{cc}8-5& 1-(-4)\\ -3-7& 2-0\end{array}\right]=\left[\begin{array}{cc}3& 5\\ -10& 2\end{array}\right]$
2. Compute $B-A$. Why this step? Swap roles; each entry flips sign because $b_{ij}-a_{ij}=-(a_{ij}-b_{ij})$. $B-A=\left[\begin{array}{cc}5-8& -4-1\\ 7-(-3)& 0-2\end{array}\right]=\left[\begin{array}{cc}-3& -5\\ 10& -2\end{array}\right]$
3. Compare. Why this step? Notice $B-A=-(A-B)$: every sign is opposite. Subtraction is anti-commutative, unlike addition.

Verify: Add the two results: $(A-B)+(B-A)$ should give $O$. $3+(-3)=0$, $5+(-5)=0$, $-10+10=0$, $2+(-2)=0$. ✓

C7 · The transpose trap (two figures)

Forecast: which one works, and does transposing "fix" the problem or change the question?

The picture below shows why $A+B$ fails: the grids don't overlap.

1. Test $A+B$. Why this step? $2\times 3$ vs $3\times 2$ — mismatch → undefined (cell C2 all over again).
2. Transpose $B$. Why this step? The transpose swaps rows and columns: $b_{ij}^{\mathsf{T}}=b_{ji}$ (see Transpose of a Matrix). So $B^{\mathsf{T}}$ becomes $2\times 3$ and the sizes now match. $B^{\mathsf{T}}=\left[\begin{array}{ccc}10& 20& 30\\ 40& 50& 60\end{array}\right]$
3. Add. Why this step? Now defined: $A+B^{\mathsf{T}}=\left[\begin{array}{ccc}1+10& 2+20& 3+30\\ 4+40& 5+50& 6+60\end{array}\right]=\left[\begin{array}{ccc}11& 22& 33\\ 44& 55& 66\end{array}\right]$

The next figure shows the transpose reshaping $B$ so its slots line up with $A$.

Verify: Entry $(2,3)$: $6+b_{23}^{\mathsf{T}}=6+b_{32}=6+60=66$. ✓

C8 · Solve for an unknown matrix

Forecast: what tool undoes "$+$ a matrix"?

1. Subtract the known matrix from both sides. Why this step? Just like numbers: to isolate $X$, add the additive inverse of the constant matrix. Every entry obeys ordinary algebra. $X=\left[\begin{array}{cc}5& 4\\ -2& 3\end{array}\right]-\left[\begin{array}{cc}2& -1\\ 0& 3\end{array}\right]$
2. Element-wise subtraction. Why this step? Valid — both $2\times 2$. $X=\left[\begin{array}{cc}5-2& 4-(-1)\\ -2-0& 3-3\end{array}\right]=\left[\begin{array}{cc}3& 5\\ -2& 0\end{array}\right]$

Verify: Plug back: $\left[\begin{array}{cc}3& 5\\ -2& 0\end{array}\right]+\left[\begin{array}{cc}2& -1\\ 0& 3\end{array}\right]=\left[\begin{array}{cc}5& 4\\ -2& 3\end{array}\right]$. ✓ Matches the right side.

C9 · Real-world word problem

Forecast: which operation is "total", which is "difference"?

1. Confirm structure matches. Why this step? Both are shop×product $2\times 2$ with identical meaning per slot — the semantic alignment the parent note demands. ✓
2. Total = $J+F$. Why this step? Combining the same relationship across two months → add. $J+F=\left[\begin{array}{cc}52& 27\\ 38& 44\end{array}\right]$
3. Difference = $F-J$. Why this step? "How many more in Feb" = February minus January, entry by entry. $F-J=\left[\begin{array}{cc}-8& 3\\ 2& -6\end{array}\right]$ Read it: Shop 1 Product 1 sold 8 fewer in February (negative = a drop), Shop 1 Product 2 sold 3 more, and so on.

Verify: Total's top-left $30+22=52$ ✓. Difference's bottom-right $19-25=-6$ ✓, and $(J+F)$ summed the same slots that $(F-J)$ differenced — consistent structure.

C10 · Exam twist — scalar-weighted combination

Forecast: do we scale first or add first? (Scale first — it distributes over each entry.)

1. Scale each matrix. Why this step? $kA$ multiplies every entry by $k$ (Scalar Multiplication of Matrices); this must happen before we combine. $2A=\left[\begin{array}{cc}4& -2\\ 8& 0\end{array}\right],3B=\left[\begin{array}{cc}3& 9\\ -6& 15\end{array}\right]$
2. Subtract element-wise. Why this step? Both scaled matrices are still $2\times 2$ → subtraction defined. $2A-3B=\left[\begin{array}{cc}4-3& -2-9\\ 8-(-6)& 0-15\end{array}\right]=\left[\begin{array}{cc}1& -11\\ 14& -15\end{array}\right]$

Verify: Slot $(2,1)$: $2(4)-3(-2)=8+6=14$. ✓ Doing the scalars and the subtraction in one shot gives the same number — that's the distributive law $2A-3B=[2a_{ij}-3b_{ij}]$.

[!recall]- Quick self-test

Connections

• Matrix operations — addition, subtraction (conditions) — the parent rule these examples drill.
• Matrix Notation and Terminology — the $a_{ij}$ slot indexing used throughout.
• Zero Matrix and Identity Matrix — cell C3's additive identity $O$.
• Scalar Multiplication of Matrices — cells C4 and C10.
• Transpose of a Matrix — cell C7's reshaping trap.
• Matrix Multiplication — contrast: it needs a different size condition.
• System of Linear Equations — the word-problem style of cell C9.