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The matrix A has x + 3 rows and x  columns. The matrix B has 4 - y rows and y + 5 columns. Both AB and BA exist. What are the values of x and y respectively?
1. 1 and 2
2. 2 and 1
3. 3 and 4
4. None of these

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Correct Answer - Option 4 : None of these

Concept: 

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 A 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  is equal to the number of rows of  matrix B,

and also a number of columns of the matrix are equal to the number of rows of the matrix 

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 × n \(\rm \begin{bmatrix} 1 & 2 & 3 \end{bmatrix}\) A matrix with one row or sometimes used to represent a vector
Column matrix × 1 \(\rm \begin{bmatrix} 4\\ 5\\ 6 \end{bmatrix}\) A matrix with one column or sometimes used to represent a vector
Square Matrix × 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.

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