If A is an m ×n matrix and B is a matrix such that both A^T B and BA^T are defined, then the ord

Name: Problem 68, Chapter 2: IIT JEE Super Course in Mathematics ALGEBRA II
Uploaded: 2021-06-24T08:51:50-08:00
Duration: 4 min 42 s
Channel: Anas Venkitta
Description: If A is an m ×n matrix and B is a matrix such that both A^T B and BA^T are defined, then the order of B is (a) m ×n (b) n ×m (c) m ×m (d) n ×n

step 1 So in the given question we are told that if the matrix A is of the order m by N, if the matrix

Question

If $\mathrm{A}$ is an $\mathrm{m} \times \mathrm{n}$ matrix and $\mathrm{B}$ is a matrix such that both $\mathrm{A}^{\mathrm{T}} \mathrm{B}$ and $\mathrm{BA}^{\mathrm{T}}$ are defined, then the order of $\mathrm{B}$ is (a) $m \times n$ (b) $\mathrm{n} \times \mathrm{m}$ (c) $\mathrm{m} \times \mathrm{m}$ (d) $n \times n$

   If $\mathrm{A}$ is an $\mathrm{m} \times \mathrm{n}$ matrix and $\mathrm{B}$ is a matrix such that both $\mathrm{A}^{\mathrm{T}} \mathrm{B}$ and $\mathrm{BA}^{\mathrm{T}}$ are defined, then the order of $\mathrm{B}$ is
(a) $m \times n$
(b) $\mathrm{n} \times \mathrm{m}$
(c) $\mathrm{m} \times \mathrm{m}$
(d) $n \times n$

IIT JEE Super Course in Mathematics ALGEBRA II

Trishna Knowledge… 2011 Edition

Chapter 2, Problem 68

Instant Answer

Solved by Expert Anas Venkitta

06/24/2021

Step 1

For matrix multiplication to be defined, the number of columns in the first matrix should be equal to the number of rows in the second matrix. Show more…

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If $\mathrm{A}$ is an $\mathrm{m} \times \mathrm{n}$ matrix and $\mathrm{B}$ is a matrix such that both $\mathrm{A}^{\mathrm{T}} \mathrm{B}$ and $\mathrm{BA}^{\mathrm{T}}$ are defined, then the order of $\mathrm{B}$ is (a) $m \times n$ (b) $\mathrm{n} \times \mathrm{m}$ (c) $\mathrm{m} \times \mathrm{m}$ (d) $n \times n$

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Key Concepts

Matrix Order (Dimensions)

The order of a matrix describes its size in terms of the number of rows and columns it contains. This concept is essential because the feasibility of many matrix operations, including multiplication, depends on the compatibility of these dimensions.

Matrix Multiplication Compatibility

Matrix multiplication is defined only when the number of columns in the first matrix equals the number of rows in the second matrix. This compatibility condition determines the dimensions of the resulting product, making it a vital concept in understanding how different matrices interact through multiplication.

Matrix Transpose

The matrix transpose is the operation that converts the rows of a matrix into its columns and vice versa. It is fundamental in many areas of linear algebra, particularly in operations involving dot products and orthogonal matrices, and plays a critical role in transforming matrices for further operations.

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