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The multiplication of matrices is quite similar to the matrix addition with a significant difference. In this matrix multiplication, we first multiply the given elements of the matrix in a certain way then we perform the usual addition among the elements.
Let's see how we multiply the elements of the two matrices.
Consider the two matrices \(A = \begin{bmatrix}
a & b\\
c & d
\end{bmatrix}\) and \(B = \begin{bmatrix}
e & f\\
g & h
\end{bmatrix}\)
a & b\\
c & d
\end{bmatrix}\) and \(B = \begin{bmatrix}
e & f\\
g & h
\end{bmatrix}\)
Using the method below, we can obtain the \(A× B\) matrix.
Then \( A × B = \begin{bmatrix}
ae + bg & af + bh\\
ce + dg& cf + dh
\end{bmatrix}\)
ae + bg & af + bh\\
ce + dg& cf + dh
\end{bmatrix}\)
Multiplication of matrices:
Matrices are multiplied by multiplying the elements in a row of the first matrix by the elements in a column of the second matrix, then adding the results.
1. To get the top leftmost entry in the product matrix, multiply the elements in the first row of the first matrix by the elements in the first column of the second matrix. Then add the product.
2. To get the second entry in the first row of the product matrix, multiply the elements in the first row of the first matrix by the elements in the second column of the second matrix and then add the product.
3. Proceeding in the same way, we can get all the entries of the product matrix.
Example:
\(\begin{bmatrix}
1 & 2\\
3 & 4
\end{bmatrix}× \begin{bmatrix}
5 & 6\\
7 & 8
\end{bmatrix} = \begin{bmatrix}
(1× 5) + (2 × 7) & (1 × 6) + ( 2 × 8)\\
(3× 5) + (4 × 7) & (3 × 6) + ( 4 × 8)
\end{bmatrix}\)
1 & 2\\
3 & 4
\end{bmatrix}× \begin{bmatrix}
5 & 6\\
7 & 8
\end{bmatrix} = \begin{bmatrix}
(1× 5) + (2 × 7) & (1 × 6) + ( 2 × 8)\\
(3× 5) + (4 × 7) & (3 × 6) + ( 4 × 8)
\end{bmatrix}\)
\(=\begin{bmatrix}
5 +14 & 6 + 16\\
15 + 28 & 18 + 32
\end{bmatrix}\begin{bmatrix}
19 & 22\\
43 & 50
\end{bmatrix}\)
5 +14 & 6 + 16\\
15 + 28 & 18 + 32
\end{bmatrix}\begin{bmatrix}
19 & 22\\
43 & 50
\end{bmatrix}\)
To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix.
Consider the multiplications of \(2×3\) and \(3×2\) matrices.
The product \(AB\) can be found only when the number of columns of matrix \(A\) is equal to the number of rows of matrix \(B\).
If the order of matrix \(A\) is \(m×n\) and \(B\) is \(n×p\) then the order of the matrix \(AB\) is \(m×p\).
Watch the below video for further clarification.
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