Properties of Matrices Transpose with Examples and Solutions

The transpose of the matrix, when it comes to Linear algebra, is among the most popular methods in matrix transformation. For a specific matrix, the transpose of a matrix is achieved by converting rows into columns or columns to rows. In the article below, you will be able to understand the definition of the term "transpose" in a matrix. We will also discuss the methods to determine the matrix's transpose and examples and properties with an explanation of the whole process.

NCERT Solutions of Class 12 Mathematics Chapter 3 Matrices

Key takeaways: Transpose of matrix, Matrix, transpose of square matrix, matrix, matrix transpose, Square matrix, columns, rows

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Properties of Transpose Of Matrix

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(i) Transpose of the Transpose Of the Matrix

To better understand this property, let’s take an example:

Let matrix be B and its transpose will be B’ or BT

Now, let’s transpose of matrix BT will be:

(BT)T = B

In this situation, the elements of B, such as,

bij will be converted into bji as a transpose of matrix B.

The elements of BT, such as,

bji will be converted into bij as a transpose of matrix BT.

(ii) Addition Property of Transpose

According to the addition property, the addition of transpose of X and Y matrices is equal to the addition of transpose of individual matrices X and Y.

See the following example to better understand:

(X+Y)' = X'+Y'

(iii) Multiplication by Constant

The matrix is multiplied by a constant and its transpose is taken to be equal to the matrix the transpose of the original matrix multiplied by that constant. Such as,

(kB)' = kB'

where k is a constant.

(iv) Multiplication Property of Transpose

According to the multiplication property, the transpose of the product of two matrices = the product of transpose of the two matrices in reverse order. Such as,

(AB)' = B'A'

The video below explains this:

Matrices Detailed Video Explanation:

Also read: Proof of the uniqueness of inverse, if it exists

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What is Matrix?

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A matrix is a collection of numbers or functions arranged in a fixed number of rows and columns in a rectangular bracket shape. Matrices have a wide range of types. To better understand the types of matrices, let's take the following examples:

A = \([ \begin{matrix} 1 & 0 & 5 \\ 3 & 2 & 4 \end{matrix} ]\)_2x3

The above matrix is of order 2 × 3. Thus, there are a total of (2 × 3 = 6) 6 elements.

Note: 2 represent the horizontal array, known as rows

And 3 represent the vertical array, known as Columns.

This matrix is known as the vertical matrix because the number of rows is less than the number of columns.

Now let’s take the another one:

A = \([ \begin{matrix} 1 & 4 \\ 3 & 6 \\ 2 & 5 \end{matrix} ]\)_3x2

This matrix is of order 2 × 3. This matrix is known as the horizontal matrix because the number of columns is less than the number of rows. Now let’s move to:

Also Read:

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What is The Transpose Of Matrix?

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The transpose of a matrix (denoted by T in the superscript of the given matrix) represents the interchange of the matrix's rows and columns or columns and rows. To better understand, let's take an example:

Suppose A is a given matrix with i × j order (i represents rows and j represents columns). The transpose will be:

​Transpose of matrix

• Order of Transpose Matrix

As mentioned above, the number of horizontal lines or rows of the matrix is denoted by "m". Furthermore, the number of vertical lines or columns is represented by "n". However, in the transpose of the matrix, m * n is exchanged into n * m. Let us take an example to better understand how m * n is exchanged into n * m in the transpose of the matrix:

Suppose a matrix:

In the above illustration, we can see that the given matrix is of order 2 × 4.

The elements of the first row [-1, 0, 4, 2] are written in the first column, and the elements in the second row [4, 8, 0, 1] are written in the second column to obtain the transpose matrix. The transpose of the matrix A is AT or A’ and has an order of 4 x 2.

Also read: Commutative law

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Transpose of a Square Matrix

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The matrix resulting from a given matrix X after altering or changing its rows to columns and columns to rows is called the transpose of a matrix X. Let us check out the transpose of the 2 × 2 square matrix to better understand the concept of square matrix.

• Transpose of a 2 × 2 Matrix

The order of a 2 × 2 matrix A will be the same while transpose. Let’s take an example:

In the same way, all the 4 × 4 or 3 × 3 matrices will be transposed into the same method.

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Things To Remember

• In reality, the transpose of a matrix means the elements of rows will be flipped to the elements of columns, and the elements of columns will be flipped into row's elements.
• The transpose is often denoted by either XT or X' or Xtr.
• If matrix A is of the order m x n, the A' or transpose of A will be n x m.
• If the matrix X has an xij element, the element will be changed into xji in the transpose of X.

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