Question:
If A is a non-zero column matrix of order m x 1 and B is non-zero row matrix of order $1 \times n$, then rank of AB is equal to
If A is a non-zero column matrix of order m x 1 and B is non-zero row matrix of order $1 \times n$, then rank of AB is equal to
Updated On: Jul 6, 2022
- m
- n
- 1
- none of these.
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The Correct Option is C
Solution and Explanation
Let
A = $\begin{bmatrix}a_{11}\\ a_{21}\\ \\ a_{m1}\end{bmatrix}$ and B = $[b_{11} \, b_{12}\,b_{13} ..... b_{1n} ]$
be two non-zero column and row matrices respectively
$\begin{bmatrix}a_{11}b_{11}&a_{11}b_{12}&a_{11}b_{13}&...&a_{11}b_{1n}\\ a_{21}b_{11}&a_{21}b_{12}&a_{21}b_{13}&...&a_{21}b_{1n}\\ .....&.....&.....&...&.....\\ a_{m1}b_{11}&a_{m1}b_{12}&a_{m1}b_{13}&...&a_{m1}b_{1n}\end{bmatrix}$
Since A, B are non-zero matrices.
$\therefore$ matrix AB will be a non-zero matrix. The matrix AB will have at least one non-zero element obtained by multiplying corresponding non-zero elements of A and B. All the two rowed minors of AB clearly vanish. Since AB is non-zero matrix,
$\therefore$ rank of AB = 1
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Concepts Used:
Matrices
Matrix:
A matrix is a rectangular array of numbers, variables, symbols, or expressions that are defined for the operations like subtraction, addition, and multiplications. The size of a matrix is determined by the number of rows and columns in the matrix.
The basic operations that can be performed on matrices are:
- Addition of Matrices - The addition of matrices addition can only be possible if the number of rows and columns of both the matrices are the same.
- Subtraction of Matrices - Matrices subtraction is also possible only if the number of rows and columns of both the matrices are the same.
- Scalar Multiplication - The product of a matrix A with any number 'c' is obtained by multiplying every entry of the matrix A by c, is called scalar multiplication.
- Multiplication of Matrices - Matrices multiplication is defined only if the number of columns in the first matrix and rows in the second matrix are equal.
- Transpose of Matrices - Interchanging of rows and columns is known as the transpose of matrices.