Lesson 6 of 8
Multiply a matrix by a number
Multiply every entry of a matrix by the same scalar.
The idea
Just like with vectors, multiplying a matrix by a scalar means multiplying every entry by that number. The size of the matrix does not change.
3A equals 3a 2 by 2 matrix, row 1: 1, 2; row 2: 0, 4 equals a 2 by 2 matrix, row 1: 3, 6; row 2: 0, 12
All four entries were multiplied by 3. The 0 stays 0, because 3 × 0 = 0.
Worked example
A equals a 2 by 2 matrix, row 1: 3, 5; row 2: 1, 6
Find 2A.
Multiply every entry
2A equals a matrix with row 1: 2 times 3, 2 times 5; row 2: 2 times 1, 2 times 6Simplify
2A equals a 2 by 2 matrix, row 1: 6, 10; row 2: 2, 12
Zero and negative scalars
Multiplying by 0 gives the zero matrix, where every entry is 0.
0A equals a 2 by 2 matrix, row 1: 0, 0; row 2: 0, 0
A negative scalar flips the sign of every entry. Zero entries stay zero.
negative 2a 2 by 2 matrix, row 1: 1, negative 3; row 2: 2, 0 equals a 2 by 2 matrix, row 1: negative 2, 6; row 2: negative 4, 0
Common mistakes
- Forgetting an entry. Check that you multiplied all of them; a 2×2 matrix has four.
- Adding the scalar. 3A means 3 × each entry, not 3 + each entry.
- Sign slips. (−2) × (−3) = 6, a positive number.