Addition, subtraction and scalar multiplication
Add or subtract matrices of the same order entry by entry. To multiply by a scalar k, multiply every entry by k: kA = [k aᵢⱼ].
Adding, subtracting and scaling
Matrices of the same order are added by adding the entries in the same positions: if A = [aᵢⱼ] and B = [bᵢⱼ] are both m × n, then A + B = [aᵢⱼ + bᵢⱼ], also m × n. Subtraction works the same way. Matrices of different orders cannot be added. A scalar is just a number. To find kA, multiply every entry of A by k. (Matrices are written row by row here, with rows separated by semicolons.)
Example 1: a combination
If A = [2, −1; 3, 0] and B = [1, 4; −2, 5], find 3A − 2B.
- Rule: kA = [k aᵢⱼ], and subtract entry by entry.
- 3A = [6, −3; 9, 0]
- 2B = [2, 8; −4, 10]
- 3A − 2B = [6 − 2, −3 − 8; 9 − (−4), 0 − 10]
3A − 2B = [4, −11; 13, −10]
Properties (A, B, C of the same order; k and l scalars)
| Property | Statement |
|---|---|
| Commutative law of addition | A + B = B + A |
| Associative law of addition | (A + B) + C = A + (B + C) |
| Additive identity | A + O = O + A = A, where O is the zero matrix of the same order |
| Additive inverse | A + (−A) = (−A) + A = O |
| Scalar times a sum of matrices | k(A + B) = kA + kB |
| Sum of scalars times a matrix | (k + l)A = kA + lA |
Example 2: two unknown matrices (board style)
Find X and Y if X + Y = [7, 3; 2, 9] and X − Y = [1, 5; 4, −1].
- Rule: matrix equations of the same order can be added and subtracted like ordinary equations.
- Add the two equations: 2X = [8, 8; 6, 8].
- So X = [4, 4; 3, 4].
- Subtract the second from the first: 2Y = [6, −2; −2, 10].
- So Y = [3, −1; −1, 5].
- Check: X + Y = [7, 3; 2, 9] and X − Y = [1, 5; 4, −1].
X = [4, 4; 3, 4], Y = [3, −1; −1, 5]
Example 3: solve for X
Find X if 2A + 3X = B, where A = [1, 2; 0, −3] and B = [8, 1; 9, −6].
- Rule: rearrange as with numbers: 3X = B − 2A.
- 2A = [2, 4; 0, −6]
- B − 2A = [8 − 2, 1 − 4; 9 − 0, −6 − (−6)] = [6, −3; 9, 0]
- X = (1/3)[6, −3; 9, 0] = [2, −1; 3, 0]
X = [2, −1; 3, 0]
A ration shop's stock matrix of rice, wheat and sugar plus the matrix of new deliveries gives the new stock by matrix addition. A 10% price rise on every item is scalar multiplication by 1.1.
1 marks: “If A + B = O, where O is the zero matrix, what is B?” B = −A, the additive inverse of A.
3 marks: “Find X and Y if 2X + Y = [4, 5; 3, 2] and X − Y = [−1, 1; −3, 1].” Add the equations: 3X = [3, 6; 0, 3], so X = [1, 2; 0, 1]. Then Y = X − [−1, 1; −3, 1] = [2, 1; 3, 0]. Check: 2X + Y = [4, 5; 3, 2].
Same order to add; a scalar hits every entry.