8.4. Matrix Algebra http://www.ck12.org
Two matrices of the same order can be subtracted by subtracting the entries in the corresponding positions.
[10 9 8
7 6 5]
−
[2 2 2
2 2 2
]
=
[8 7 6
5 4 3
]
You can find the product of matrixAand matrixBif the number of columns in matrixAmatches the number of rows
in matrixB. Another way to remember this is when you write the orders of matrixAand matrixBnext to each other
they must be connected by the same number. The resulting matrix has the number of rows from the first matrix and
the number of columns from the second matrix.
( 2 × 3 )·( 3 × 5 ) = ( 2 × 5 )
To compute the first entry of the resulting 2×5 matrix you should match the first row from the first matrix and the
first column of the second matrix. The arithmetic operation to combine these numbers is identical to taking the dot
product between two vectors.
- The entry in the first row first column of the new matrix is computed as 1· 0 + 4 · 2 + 3 · 1 =11.
- The entry in the second row first column of the new matrix is computed as 5· 0 + 6 · 2 + 9 · 1 =21.
- The rest of the entries of this product are left to Example A.
PropertiesofMatrixAlgebra
- Commutatively holds for matrix addition. This means that when matricesAandBcan be added (when they
have matching orders), then:A+B=B+A - Commutativelydoes nothold in general for matrix multiplication.
- Associativity holds for both multiplication and addition.(AB)C=A(BC),(A+B)+C=A+(B+C)
- Distribution over addition and subtraction holds.A(B±C) =AB±AC
Example A
Complete the entries of the matrix multiplication introduced in the guidance section.
[1 4 3
5 6 9]
·
0 1 3 1 0
2 0 0 2 1
1 1 3 0 1
Solution: Two of the arithmetic operations are shown.
C=
[ 11 4 12 9 7
21 14 42 17 15
]
c 12 = 1 · 1 + 4 · 0 + 3 · 1 = 4
c 22 = 5 · 1 + 6 · 0 + 9 · 1 = 14Example B
Show the commutative property does not hold by demonstratingAB 6 =BA