Question

$If \:A\: = \: \begin{gathered}\left[\begin{array}{ccc}3&2 \\ \\ - 1&3 \\ \\ 7&9\end{array}\right]\end{gathered}_{3 \times 2}$

$Find\:A^T$
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Answer 1

Transpose - Matrices

Basic Knowledge:

The matrix in mathematics is defined as an array or a string of elements. The elements can be numbers, symbols or alphabets.

A matrix consists of a number of rows and columns.

A matrix is represented as:

$\left[\begin{array}{cc}a & b \\ c &d\end{array}\right]_{m \times n}$

Here,

• m × n - the number of elements.
• m - the number of rows.
• n - the number of columns.

To find the transpose of matrix we simply interchange row and columns.

The transpose of a matrix can be found using:

$\longrightarrow m \times n = n \times m$

Means the first row will become the first column and the second row will become the second column of the transpose. We can also say that, the rows of the matrix becomes the columns of its transpose.

The given matrix is,

$\longrightarrow A = \left[\begin{array}{cc}3&2 \\ - 1&3 \\ 7&9\end{array}\right]_{3 \times 2}$

It has three rows and two columns, we need to find the transpose of the given matrix. It is written as $A^T$.

Solution:

The transpose of a matrix is given by interchanging rows and columns.

$\left[\begin{array}{cc}3&2 \\ - 1&3 \\ 7&9\end{array}\right]_{3 \times 2}$

The first row becomes the first column, the first row is 3, 2, so the first column of the transpose will be 3, 2. The second row becomes the second column, the second row is -1, 3, so the second column of the transpose will be -1, 3. The third row becomes the third column, the third row is 7, 9, so the third column of the transpose will be 7, 9.

$\implies \left[\begin{array}{ccc} 3 & -1 & 7 \\2 & 3 & 9\end{array}\right]$

Therefore our final answer is:

$\boxed{A^T = \left[\begin{array}{ccc} 3 & -1 & 7 \\2 & 3 & 9\end{array}\right]}$

$\rule{300}{2}$

Conclusion:

The matrix has three rows and two columns.

The first row became the first column. The second row became the second column. And the third row became the third column.

The rows of the matrix became the columns or we can also say that the columns of the matrix became the rows of the matrix of its transpose.

C1 [column 1] became the R1 [row 1] and C2 [column 2] became the R2 [row 2].

The matrix A has 3 rows and 2 columns its order will be $3 \times 2$. And the A transpose has 2 rows and 3 columns, its order will be $2 \times 3$.

The order is also reversed. $3 \times 2$ becomes $2 \times 3$.

$\longrightarrow 3 \times 2 = 2 \times 3$