Matrix Multiplication: Definition, Formulas and Examples,

Matrix multiplication is a crucial operation in linear algebra, where two matrices are multiplied to produce a third matrix. This process involves taking the dot product of rows from the first matrix with columns from the second matrix.

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1. Introduction to the Matrix Multiplication:

Matrix multiplication is a fundamental operation in linear algebra, essential for solving complex mathematical problems and practical applications in various fields, including computer graphics, engineering, and data science. This guide aims to provide a detailed understanding of matrix multiplication, its rules, formulas, and applications.

2. What is Matrix Multiplication:

Matrix multiplication involves multiplying two matrices to produce a third matrix. Unlike simple arithmetic multiplication, matrix multiplication combines the rows of the first matrix with the columns of the second matrix to produce the resultant matrix.

3. How to multiply two Matrices:

To multiply two matrices, follow these steps:

• Ensure the number of columns in the first matrix equals the number of rows in the second matrix.
• Multiply corresponding elements and sum them up to get the elements of the resultant matrix.

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4. Rules for Matrix Multiplication:

• Order Rule: Matrix A (of order m × n) can be multiplied by matrix B (of order n × p).
• Associativity: (AB)C = A(BC)
• Distributivity: A(B C) = AB AC
• Non-Commutativity: AB $\ne$ BA generally
• Matrix multiplication indicates rows by columns multiplication.

5. 2x2 Matrix Multiplication Formula:

For two 2x2 matrices A and B: A = $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ and B = $\begin{bmatrix} e & f \\ g & h \end{bmatrix}$

The product C = AB = $\begin{bmatrix} ae bg & af bh \\ ce dg & cf dh \end{bmatrix}$

6. 3x3 Matrix Multiplication Formula

For two 3x3 matrices A and B: A = $\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}$ and B = $\begin{bmatrix} j & k & l \\ m & n & o \\ p & q & r \end{bmatrix}$

The product C = AB = $\begin{bmatrix} aj bm cp & ak bn cq & al bo cr \\ dj em fp & dk en fq & dl eo fr \\ gj hm IP & gk hn iq & gl ho ir \end{bmatrix}$

7. Properties of Matrix Multiplication:

8. Matrix Multiplication Solved Examples:

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

Solution: AB = $\begin{bmatrix} 1.5 2.7 & 1.6 2.8 \\ 3.5 4.7 & 3.6 4.8 \end{bmatrix}$ = $\begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}$

Example 2: Find PQ if P = $\begin{bmatrix} 2 & 4 & 6 \\ 1 & 3 & 5 \end{bmatrix}$ and Q = $\begin{bmatrix} 7 & 8 \\ 9 & 10 \\ 11 & 12 \end{bmatrix}$

Solution: PQ = $\begin{bmatrix} 2.7 4.9 6.11 & 2.8 4.10 6.12 \\ 1.7 3.9 5.11 & 1.8 3.10 5.12 \end{bmatrix}$ = $\begin{bmatrix} 106 & 128 \\ 89 & 98 \end{bmatrix}$

9. Practice questions on Matrix Multiplication:

Question 1: Find AB and BA if A = $\begin{bmatrix} 1 & 0 & 2 \\ 0 & 1 & 3 \end{bmatrix}$ and B = $\begin{bmatrix} 3 & 4 \\ 1 & 2 \\ 5 & 6 \end{bmatrix}$

Question 2: Find PP, PQ, QP, QQ if Q = $\begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{bmatrix}$ and P = $\begin{bmatrix} 7 & -8 \\ 9 & 10 \\ \end{bmatrix}$

10. FAQs on Matrix Multiplication:

What is Matrix Vector Multiplication?

Matrix vector multiplication involves multiplying a matrix by a vector, resulting in a new vector. It's commonly used in transformations and linear equations.

Is Matrix Multiplication Always Defined?

No, matrix multiplication is only defined when the number of columns in the first matrix equals the number of rows in the second matrix.

What is the Product of Multiplication of Matrices of Orders 2×1 and 2×2?

Matrix multiplication is not possible because the number of columns in the first matrix (1) does not equal the number of rows in the second matrix (2).

Is Matrix Multiplication Always Commutative?

No, matrix multiplication is generally not commutative. AB ≠ BA in most cases.

When is Matrix Multiplication Possible?

Matrix multiplication is possible when the number of columns in the first matrix equals the number of rows in the second matrix.

Can You Multiply Matrices of Order 3x1 and 2x2?

No, because the number of columns in the first matrix (1) does not equal the number of rows in the second matrix (2).

11. Real-life application of Matrix Multiplication:

Matrix multiplication is widely used in various fields, including:

• Computer Graphics: Transforming coordinates and modeling 3D objects.
• Engineering: Solving systems of linear equations.
• Economics: Input-output models for economic analysis.
• Data Science: Machine learning algorithms and data transformations.

12. Conclusion:

Matrix multiplication is a powerful mathematical tool, and its applications span numerous fields. Understanding matrix multiplication's rules, formulas, and properties allows for solving complex problems and implementing advanced algorithms.

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