19.10 - Transpose of a Matrix

In the previous section, we completed a discussion on multiplication of matrices. In this section, we will see transpose of a matrix.

Transpose of a Matrix

This can be written in 9 steps:
1. Suppose that, we are given a matrix A.
• Based on A, we can write a new matrix A' in such a way that:
    ♦ Rows in A, are the columns in A'.
    ♦ Columns in A, are the rows in A'.
2. In other words,
    ♦ First row of A, becomes the first column in A'.   
    ♦ Second row of A, becomes the second column in A'.   
    ♦ Third row of A, becomes the third column in A'.   
    ♦ so on . . .
3. This is same as:
    ♦ First column of A, becomes the first row in A'.   
    ♦ Second column of A, becomes the second row in A'.   
    ♦ Third column of A, becomes the third row in A'.
    ♦ so on . . .
4. Let us see an example:
If A = $\left[\begin{array}{r}               
-7    &{    0    }    \\
8    &{    \sqrt{2}    }    \\
-{\frac{1}{3}}    &{    3    }    \\
\end{array}\right]               
$, then A' = $\left[\begin{array}{r}                                       
-7    &{    8    }    &{    -{\frac{1}{3}}    }       \\
0    &{    \sqrt{2}    }    &{    3    }        \\
\end{array}\right]                                       
$
5. Consider any element, say $-{\frac{1}{3}}$ in A. It’s position is: row 3, column 1.
• But in A', the position of $-{\frac{1}{3}}$ is: row 1, column 3.
    ♦ That is., the row and column are interchanged.
6. This interchanging of rows and columns is applicable to all elements in general.
• We can write:
    ♦ In the original matrix, an element is at position i,j.
    ♦ Then in the new matrix, that element will be at the position j,i.
7. Also note that:
If the order of the original matrix is m×n, then the order of the new matrix will be n×m.
8. Based on the above steps, we can write the definition:
If A = [a_ij]_m×n, then A' = [a_ji]_n×m .
9. The new matrix A' is called the transpose of A. It can be denoted either as A' or A^T.

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Properties of transpose of matrices

• We have to be familiar with four properties:
For any matrices A and B of suitable orders, we have
I. (A')' = A                II. (kA)' = kA' (where k is any constant)
III. (A+B)' = A' + B'        IV. (AB)' = B' A'.

[We mention ‘suitable orders’ because, any two given matrices cannot be added or multiplied. If we want (A+B), then A and B must be of the same order. Similarly, if we want (AB), then number of columns in A must be equal to the number of rows in B]

• We will see the proofs of the four properties in higher classes. At present, we will verify them using examples.
Let A = $\left[\begin{array}{r}                       
9    &{    6    }    &{    -4    }\\
0    &{    \sqrt{2}    }    &{    1    }\\
\end{array}\right]                       
$ and B = $\left[\begin{array}{r}                       
-1    &{    2    }    &{    5    }\\
11    &{    7    }    &{    3    }\\
\end{array}\right] $.

1. Verifying property I:

2. Verifying property II:

• From (4) and (6), we get:
(kB)' = kB' (where k is any constant) 

3. Verifying property III:

• From (5) and (7), we get:
(A+B)' = A' + B'

4. Verifying property IV:
Let X = $\left[\begin{array}{r}                       
9    &{    6    }    &{    -4    }\\
0    &{    {2}    }    &{    1    }\\
\end{array}\right]                       
$ and Y = $\left[\begin{array}{r}                       
-1    &{    2    }    &{    5    }\\
11    &{    7    }    &{    3    }\\
7    &{    1    }    &{    4    }\\
\end{array}\right] $.

• From (3) and (7), we get:
(XY)' = Y' X'
(note that, positions of X and Y are interchanged)

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In the next section, we will see symmetric and skew symmetric matrices.

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