Transpose of a Matrix: Overview, Questions, Preparation

A network is a rectangular exhibit of numbers or capacities orchestrated in a fixed number of lines and segments. 

There are numerous kinds of matrices.

The above lattice An is of order 3 × 2. In this manner, there is a sum of 6 components. 

The event exhibit is known as lines, and the vertical cluster is known as Columns. 
The quantity of lattice An is more prominent than the number of segments; such a grid is known as a Vertical framework.    

The quantity of segments in framework B is more prominent than the number of columns. Such a lattice is known as a Horizontal grid.

A lattice P is supposed to be equivalent to framework Q if their requests are equivalent, and each related component of P is equivalent to that of Q. 

Transpose

The quantity of lines and segments in An is equivalent to the number of sections and columns in B individually. Hence, network B is known as the Transpose of framework A. The translation of framework An is spoken by A′ or AT. The accompanying assertion sums up a translation of a framework: 

If A = [aij]m×n, at that point A′ =[aij]n×m. 

Source: NCERT

If we take a translation of the rendered grid, the lattice is equivalent to the first network. Henceforth, for a network A, 

(A′)′ = A 

What fundamentally occurs is that any component of A, for example, aij, gets changed over to aji if An's translation is taken. Along these lines, taking translate once more, it gets changed over to aij, the first grid A.

Multiplication property of transpose
Translation of the result of two frameworks is equivalent to the render of the two networks backward request. That is 

(AB)′ = B′A′

Addition Property of Transpose 

Matrix id Render of an expansion of two grids An and B got will be equivalent to the amount of translation of individual network An and B. 

This implies, 

(A+B)′ = A′+B′

Multiplication by constant

On the off chance that a lattice is increased by a steady and its translation is taken, at that point, the network acquired is equivalent to rendering a unique grid duplicated by that consistency. That is, 

(kA)′ = kA′, where k is a steady

In class 11, the topic matrices have been discussed with different questions and answers. It has a weightage of 13 Marks.

1. Suppose the order of A is mxn. Also, the order for B is p×n. At that point, the order for matrix AB is?

Solution.

AB is defined only when the order of B is n×x

BA is defined only when the order of B is n×m,i.e.,x=m

∴ order of matrix B is m×n

2. Translate of a rectangular matrix is a

Solution.

As it will be in a rectangular form of presentation, its matrix will be in tabular Rectangular matrix form.

3. In matrices (AB)⁻¹ equals to
Solution.

A − 1 is invertible, and its inverse is ( A − 1 ) − 1 = A . AB is invertible, and its inverse is ( AB ) − 1 = B − 1 A − 1.

Q: Where can we use the transpose of a matrix?

Q: How can we use a nonsquare matrix?

Q: What is the matrix conjugate?

Q: Is symmetric transpose?

Q: Inverse and transpose are the same?