How to do multiplication of matrix 3×3

Matrix Multiplication – Complete Method for Grade 12 Students

Matrix multiplication is an important concept in Class 12 Mathematics. It is used in solving systems of equations, transformations, and many real-life applications. This guide explains the complete step-by-step method of multiplying matrices with clear examples.

1. Condition for Matrix Multiplication

Two matrices can be multiplied only when:

Number of columns of first matrix = Number of rows of second matrix

If matrix $A$ is of order $m \times n$ and matrix $B$ is of order $n \times p$, then product matrix $AB$ will be of order $m \times p$.

Example: $A_{2 \times 2} \times B_{2 \times 2}$ is possible $A_{2 \times 3} \times B_{3 \times 2}$ is possible $A_{2 \times 2} \times B_{3 \times 2}$ is not possible

2 × 2 Matrix Multiplication

Method

Step 1: Take first row of first matrix.
Step 2: Take first column of second matrix.
Step 3: Multiply corresponding elements.
Step 4: Add the products.
Step 5: Write the result in first position.
Step 6: Repeat for all rows and columns.

Example 1

Multiply the matrices:

$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$

Solution

First row of A × First column of B

$(1 \times 5) + (2 \times 7) = 5 + 14 = 19$

First row of A × Second column of B

$(1 \times 6) + (2 \times 8) = 6 + 16 = 22$

Second row of A × First column of B

$(3 \times 5) + (4 \times 7) = 15 + 28 = 43$

Second row of A × Second column of B

$(3 \times 6) + (4 \times 8) = 18 + 32 = 50$

Final Answer:

$AB = \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}$

Example 2

Multiply:

$A = \begin{bmatrix} 2 & 0 \\ 1 & 3 \end{bmatrix}$ $B = \begin{bmatrix} 4 & 5 \\ 6 & 7 \end{bmatrix}$

Solution

$(2 \times 4) + (0 \times 6) = 8$

$(2 \times 5) + (0 \times 7) = 10$

$(1 \times 4) + (3 \times 6) = 4 + 18 = 22$

$(1 \times 5) + (3 \times 7) = 5 + 21 = 26$

Final matrix:

$AB = \begin{bmatrix} 8 & 10 \\ 22 & 26 \end{bmatrix}$

3 × 3 Matrix Multiplication

Method

Step 1: Multiply row of first matrix with column of second matrix.
Step 2: Multiply corresponding elements.
Step 3: Add all products.
Step 4: Write result in position.
Step 5: Continue for all rows and columns.

Example 3

Multiply:

$A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}$

$B = \begin{bmatrix} 1 & 0 & 2 \\ 0 & 1 & 2 \\ 1 & 1 & 0 \end{bmatrix}$

Solution

First row × First column

$(1×1)+(2×0)+(3×1) = 1 + 0 + 3 = 4$

First row × Second column

$(1×0)+(2×1)+(3×1) = 0 + 2 + 3 = 5$

First row × Third column

$(1×2)+(2×2)+(3×0) = 2 + 4 = 6$

Second row × First column

$(4×1)+(5×0)+(6×1) = 4 + 6 = 10$

Second row × Second column

$(4×0)+(5×1)+(6×1) = 5 + 6 = 11$

Second row × Third column

$(4×2)+(5×2)+(6×0) = 8 + 10 = 18$

Third row × First column

$(7×1)+(8×0)+(9×1) = 7 + 9 = 16$

Third row × Second column

$(7×0)+(8×1)+(9×1) = 8 + 9 = 17$

Third row × Third column

$(7×2)+(8×2)+(9×0) = 14 + 16 = 30$

Final Matrix:

$AB = \begin{bmatrix} 4 & 5 & 6 \\ 10 & 11 & 18 \\ 16 & 17 & 30 \end{bmatrix}$

Example 4

Multiply:

$A = \begin{bmatrix} 2 & 1 & 0 \\ 3 & 2 & 1 \\ 1 & 0 & 2 \end{bmatrix}$

$B = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{bmatrix}$

Solution

First row × First column

$(2×1)+(1×0)+(0×5) = 2$

First row × Second column

$(2×2)+(1×1)+(0×6) = 4 + 1 = 5$

First row × Third column

$(2×3)+(1×4)+(0×0) = 6 + 4 = 10$

Second row × First column

$(3×1)+(2×0)+(1×5) = 3 + 5 = 8$

Second row × Second column

$(3×2)+(2×1)+(1×6) = 6 + 2 + 6 = 14$

Second row × Third column

$(3×3)+(2×4)+(1×0) = 9 + 8 = 17$

Third row × First column

$(1×1)+(0×0)+(2×5) = 1 + 10 = 11$

Third row × Second column

$(1×2)+(0×1)+(2×6) = 2 + 12 = 14$

Third row × Third column

$(1×3)+(0×4)+(2×0) = 3$

Final Matrix:

$AB = \begin{bmatrix} 2 & 5 & 10 \\ 8 & 14 & 17 \\ 11 & 14 & 3 \end{bmatrix}$

Important Points for Students

1. Matrix multiplication is not commutative, that is $AB \ne BA$.
2. Always check order before multiplication.
3. Multiply row by column only.
4. Add all products carefully.
5. Practice more to improve speed.

Conclusion

Matrix multiplication becomes easy when students follow the row by column rule carefully. Regular practice of 2×2 and 3×3 matrices helps in mastering this concept for board exams and competitive exams.