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Let'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 \((AB)C = A (BC)\).
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.
\((AB) C= \left ( \begin{bmatrix}
1 & 2 \\
3 & 4
\end{bmatrix} \begin{bmatrix}
5 & 6\\
7 & 8
\end{bmatrix} \right )\begin{bmatrix}
9 & 10\\
11 & 12
\end{bmatrix}\)
1 & 2 \\
3 & 4
\end{bmatrix} \begin{bmatrix}
5 & 6\\
7 & 8
\end{bmatrix} \right )\begin{bmatrix}
9 & 10\\
11 & 12
\end{bmatrix}\)
\(=\begin{bmatrix}
(1×5) + (2×7) & (1×6) + (2×8)\\
(3×5) + (4×7) & (3×6) + (4×8)
\end{bmatrix} \begin{bmatrix}
9 & 10\\
11 & 12
\end{bmatrix}\)
(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}\)
\(=\begin{bmatrix}
5 + 14 & 6 + 18\\
15+28 & 18+32
\end{bmatrix} \begin{bmatrix}
9 & 10\\
11 & 12
\end{bmatrix}\)
5 + 14 & 6 + 18\\
15+28 & 18+32
\end{bmatrix} \begin{bmatrix}
9 & 10\\
11 & 12
\end{bmatrix}\)
\(=\begin{bmatrix}
19 & 24\\
43 & 50
\end{bmatrix} \begin{bmatrix}
9 & 10\\
11 & 12
\end{bmatrix}\)
19 & 24\\
43 & 50
\end{bmatrix} \begin{bmatrix}
9 & 10\\
11 & 12
\end{bmatrix}\)
\(=\begin{bmatrix}
(19 × 9) + (24×11)& (19 × 10) + (24×12)\\
(43 × 9) + (50×11)& (43 × 10) + (50×12)
\end{bmatrix}\)
(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)
171+264 & 190 + 288\\
387 + 550 & 430 + 600
\end{bmatrix}=\begin{bmatrix}
435 & 478\\
937 & 1030
\end{bmatrix}\)……….(1)
Similarly, let's find \(A(BC)\).
\((AB) C= \begin{bmatrix}
1 & 2 \\
3 & 4
\end{bmatrix} \left (\begin{bmatrix}
5 & 6\\
7 & 8
\end{bmatrix} \begin{bmatrix}
9 & 10\\
11 & 12
\end{bmatrix}\right )\)
1 & 2 \\
3 & 4
\end{bmatrix} \left (\begin{bmatrix}
5 & 6\\
7 & 8
\end{bmatrix} \begin{bmatrix}
9 & 10\\
11 & 12
\end{bmatrix}\right )\)
\(A( B C) =\begin{bmatrix}
1 & 2\\
3 & 4
\end{bmatrix} \begin{bmatrix}
(5×9)+ (6 × 11) & (5×10)+ (6 × 12)\\
(7×9)+ (8 × 11) & (7×10)+ (8 × 12)
\end{bmatrix}\)
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}\)
\( =\begin{bmatrix}
1& 2\\
3& 4
\end{bmatrix} \begin{bmatrix}
111 & 122\\
151 & 166
\end{bmatrix}\)
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)
(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.
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