UPSKILL MATH PLUS
Learn Mathematics through our AI based learning portal with the support of our Academic Experts!
Learn moreLet's learn the various properties of multiplication of matrix as follows:
1. Commutative property
2. Associative property
3. Distributive property
4. Multiplication of a matrix by a unit matrix
Let's dive into each property individually with an example.
Commutative property of matrix multiplication:
In general, matrix multiplication is not commutative.
If A is of order m×n and B of the order n×p then AB is defined but BA is not defined. Even if AB and BA are both defined, they don't need to be equal.
Therefore, in general AB ≠ BA.
Associative property:
Matrix multiplication always satisfies the associative property.
That is (AB)C = A(BC).
Example:
If A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, B = \begin{bmatrix} 5 & 6\\ 7 & 8 \end{bmatrix}, C = \begin{bmatrix} 9 & 10\\ 11 & 12 \end{bmatrix} then verify that (AB)C = A (BC).
Solution:
First, we find the sum of AB matrices, then multiply its result with the \(C) matrix.
(AB) C= \left ( \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} 5 & 6\\ 7 & 8 \end{bmatrix} \right )\begin{bmatrix} 9 & 10\\ 11 & 12 \end{bmatrix}
=\begin{bmatrix} (1×5) + (2×7) & (1×6) + (2×8)\\ (3×5) + (4×7) & (3×6) + (4×8) \end{bmatrix} \begin{bmatrix} 9 & 10\\ 11 & 12 \end{bmatrix}
=\begin{bmatrix} 5 + 14 & 6 + 18\\ 15+28 & 18+32 \end{bmatrix} \begin{bmatrix} 9 & 10\\ 11 & 12 \end{bmatrix}
=\begin{bmatrix} 19 & 24\\ 43 & 50 \end{bmatrix} \begin{bmatrix} 9 & 10\\ 11 & 12 \end{bmatrix}
=\begin{bmatrix} (19 × 9) + (24×11)& (19 × 10) + (24×12)\\ (43 × 9) + (50×11)& (43 × 10) + (50×12) \end{bmatrix}
= \begin{bmatrix} 171+264 & 190 + 288\\ 387 + 550 & 430 + 600 \end{bmatrix}=\begin{bmatrix} 435 & 478\\ 937 & 1030 \end{bmatrix}……….(1)
Similarly, let's find A(BC).
(AB) C= \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \left (\begin{bmatrix} 5 & 6\\ 7 & 8 \end{bmatrix} \begin{bmatrix} 9 & 10\\ 11 & 12 \end{bmatrix}\right )
A( B C) =\begin{bmatrix} 1 & 2\\ 3 & 4 \end{bmatrix} \begin{bmatrix} (5×9)+ (6 × 11) & (5×10)+ (6 × 12)\\ (7×9)+ (8 × 11) & (7×10)+ (8 × 12) \end{bmatrix}
=\begin{bmatrix} 1& 2\\ 3& 4 \end{bmatrix} \begin{bmatrix} 111 & 122\\ 151 & 166 \end{bmatrix}
= \begin{bmatrix} (1×111)+ (2× 151) & (1×122)+ (2 × 166)\\ (3×111)+ (4× 151) & (3×122)+ (4× 166) \end{bmatrix}=\begin{bmatrix} 413 & 454\\ 937 & 1030 \end{bmatrix}……….(2)
From (1), (2) we can see that (AB)C) = A(BC).
Thus, the given matrix satisfies the associative property of matrices.
Register for free to see more content