Matrix Multiplication – Definition, Conditions, Examples

Matrix Multiplication

Matrix multiplication is an operation where two matrices are multiplied to produce a new matrix. Unlike addition and subtraction, matrix multiplication follows specific rules and is not performed element-wise.

For two matrices $A$ and $B$ to be multiplied:

• The number of columns in the first matrix $A$ must be equal to the number of rows in the second matrix $B$.
• If $A$ is of size $m\times n$ and $B$ is of size $n\times p$, then their product $C=A\times B$ will be of size $m\times p$.
• The element at position $C_{ij}$ in the resulting matrix is computed as the dot product of the $i^{th}$ row of $A$ and the $j^{th}$ column of $B$.

Mathematically, matrix multiplication is expressed as:

$$C_{ij}=\sum _{k=1}^n{A}_{ik}\cdot B_{kj}$$

Example 1: Multiplying a 2×3 Matrix with a 3×2 Matrix

Consider the following matrices:

$$A=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right]$$

$$B=\left[\begin{array}{cc}7& 8\\ 9& 10\\ 11& 12\end{array}\right]$$

Matrix $A$ has 2 rows and 3 columns, and matrix $B$ has 3 rows and 2 columns. Since the number of columns in $A$ matches the number of rows in $B$, we can multiply them.

The result $C=A\times B$ will be a $2\times 2$ matrix.

Step 1: Compute $C_{11}$ (First Row, First Column)

The element $C_{11}$ is obtained by multiplying the first row of $A$ with the first column of $B$ and summing the products:

$$C_{11}=(1\times 7)+(2\times 9)+(3\times 11)$$

$$C_{11}=7+18+33=58$$

Step 2: Compute $C_{12}$ (First Row, Second Column)

The element $C_{12}$ is obtained by multiplying the first row of $A$ with the second column of $B$ and summing the products:

$$C_{12}=(1\times 8)+(2\times 10)+(3\times 12)$$

$$C_{12}=8+20+36=64$$

Step 3: Compute $C_{21}$ (Second Row, First Column)

The element $C_{21}$ is obtained by multiplying the second row of $A$ with the first column of $B$ and summing the products:

$$C_{21}=(4\times 7)+(5\times 9)+(6\times 11)$$

$$C_{21}=28+45+66=139$$

Step 4: Compute $C_{22}$ (Second Row, Second Column)

The element $C_{22}$ is obtained by multiplying the second row of $A$ with the second column of $B$ and summing the products:

$$C_{22}=(4\times 8)+(5\times 10)+(6\times 12)$$

$$C_{22}=32+50+72=154$$

Final Result

Thus, the product of $A$ and $B$ is:

$$C=\left[\begin{array}{cc}58& 64\\ 139& 154\end{array}\right]$$

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Example 2: Multiplication of Two 2×2 Matrices

Consider the two 2×2 matrices:

$$A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right],B=\left[\begin{array}{cc}5& 6\\ 7& 8\end{array}\right]$$

Since $A$ is a $2\times 2$ matrix and $B$ is also a $2\times 2$ matrix, their product will be a $2\times 2$ matrix.

Step 1: Compute $C_{11}$ (First Row, First Column)

The element $C_{11}$ is found by taking the dot product of the first row of $A$ and the first column of $B$:

$$C_{11}=(1\times 5)+(2\times 7)$$

– Multiply the first element of row 1 of $A$ by the first element of column 1 of $B$: $1\times 5=5$

– Multiply the second element of row 1 of $A$ by the second element of column 1 of $B$: $2\times 7=14$

– Add these values: $5+14=19$.

Step 2: Compute $C_{12}$ (First Row, Second Column)

The element $C_{12}$ is found by taking the dot product of the first row of $A$ and the second column of $B$:

$$C_{12}=(1\times 6)+(2\times 8)$$

– Multiply the first element of row 1 of $A$ by the first element of column 2 of $B$: $1\times 6=6$

– Multiply the second element of row 1 of $A$ by the second element of column 2 of $B$: $2\times 8=16$

– Add these values: $6+16=22$.

Step 3: Compute $C_{21}$ (Second Row, First Column)

The element $C_{21}$ is found by taking the dot product of the second row of $A$ and the first column of $B$:

$$C_{21}=(3\times 5)+(4\times 7)$$

– Multiply the first element of row 2 of $A$ by the first element of column 1 of $B$: $3\times 5=15$

– Multiply the second element of row 2 of $A$ by the second element of column 1 of $B$: $4\times 7=28$

– Add these values: $15+28=43$.

Step 4: Compute $C_{22}$ (Second Row, Second Column)

The element $C_{22}$ is found by taking the dot product of the second row of $A$ and the second column of $B$:

$$C_{22}=(3\times 6)+(4\times 8)$$

– Multiply the first element of row 2 of $A$ by the first element of column 2 of $B$: $3\times 6=18$

– Multiply the second element of row 2 of $A$ by the second element of column 2 of $B$: $4\times 8=32$

– Add these values: $18+32=50$.

Final Result

The resulting 2×2 matrix is:

$$C=\left[\begin{array}{cc}19& 22\\ 43& 50\end{array}\right]$$

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Example 3: Multiplication of Two 3×3 Matrices

Consider the two 3×3 matrices:

$$A=\left[\begin{array}{ccc}2& 3& 4\\ 1& 0& 5\\ 7& 6& 8\end{array}\right],B=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]$$

The result $C=A\times B$ will be a 3×3 matrix. We compute each element as follows:

Step 1: Compute First Row

$$C_{11}=(2\times 1)+(3\times 4)+(4\times 7)=2+12+28=42$$

$$C_{12}=(2\times 2)+(3\times 5)+(4\times 8)=4+15+32=51$$

$$C_{13}=(2\times 3)+(3\times 6)+(4\times 9)=6+18+36=60$$

Step 2: Compute Second Row

$$C_{21}=(1\times 1)+(0\times 4)+(5\times 7)=1+0+35=36$$

$$C_{22}=(1\times 2)+(0\times 5)+(5\times 8)=2+0+40=42$$

$$C_{23}=(1\times 3)+(0\times 6)+(5\times 9)=3+0+45=48$$

Step 3: Compute Third Row

$$C_{31}=(7\times 1)+(6\times 4)+(8\times 7)=7+24+56=87$$

$$C_{32}=(7\times 2)+(6\times 5)+(8\times 8)=14+30+64=108$$

$$C_{33}=(7\times 3)+(6\times 6)+(8\times 9)=21+36+72=129$$

Final Result

The resulting 3×3 matrix is:

$$C=\left[\begin{array}{ccc}42& 51& 60\\ 36& 42& 48\\ 87& 108& 129\end{array}\right]$$

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Conclusion

Matrix multiplication is a fundamental operation in linear algebra used in various applications such as computer graphics, machine learning, and engineering. The key points to remember are:

• Matrix multiplication is not commutative ($A\times B\ne B\times A$ in general).
• The number of columns in the first matrix must match the number of rows in the second matrix.
• Each element of the resulting matrix is computed as the dot product of a row from the first matrix and a column from the second matrix.

By following these principles, you can confidently perform matrix multiplication step by step.