Correct Answer - Option 4 : None of these
Concept:
A matrix is a rectangular arrangement of numbers into rows and columns. Matrices are commonly written in rectangular brackets.
The horizontal and vertical lines of entries in a matrix are called rows and columns, respectively.
The size of a matrix is defined by the number of rows and columns that it contains.
A matrix with m rows and n columns is called an m × n matrix or m by n matrix, while m and n are called its dimensions.
For example, matrix AAA has two rows and three columns.
\(\rm \begin{bmatrix} -1 & 2 & 3\\ 4 & 5 & -6 \end{bmatrix}\)
so, matrix A is 2 × 3 matrix.
Matrix Identities:
- If A is an n × m matrix and B is an m × p matrix, the result AB of their multiplication is an n ×p matrix defined only if the number of columns m in A is equal to the number of rows m in B.
- If A, B are, respectively m × n, k × l matrices, then both AB and BA are defined if and only if n = k and l = m.
Calculation:
Given matrix A has x + 3 rows and x columns. The matrix B has 4 - y rows and y + 5 columns and also AB and BA exist,
The number of columns of the matrix A is equal to the number of rows of matrix B,
x = 4 - y
and also a number of columns of the matrix are equal to the number of rows of the matrix A
y + 5 = x + 3
Solving both linear equations, we get
x = 3 and y = 1
Application:
Matrices or matrix is commonly used its mathematics, also matrix has various uses like:
- Encryption
- Games especially 3D
- Economics and business
- Construction
- Dance – contra dance
- Animation
- Physics
- Geology
Name |
Size |
Example |
Description |
Row matrix |
1 × n |
\(\rm \begin{bmatrix} 1 & 2 & 3 \end{bmatrix}\) |
A matrix with one row or sometimes used to represent a vector |
Column matrix |
n × 1 |
\(\rm \begin{bmatrix} 4\\ 5\\ 6 \end{bmatrix}\) |
A matrix with one column or sometimes used to represent a vector |
Square Matrix |
n × n |
\(\rm \begin{bmatrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9 \end{bmatrix}\) |
A matrix with the same number of rows and columns or sometimes used to represent a linear transformation from a vector space to itself, such as reflection, rotation, or shearing.
|
Note:
- If AB is defined, then BA need not be defined.
- If A, B are, respectively m × n, k × l matrices, then both AB and BA are defined if and only if n = k and l = m.
- If AB and BA are both defined, it is not necessary that AB = BA.
- If the product of two matrices is a zero matrix, it is not necessary that one of the matrices is a zero matrix.
- For three matrices A, B and C of the same order, if A = B, then AC = BC, but the converse is not true.
- A. A = A2, A. A. A = A3, so on.