Solved Examples on Column Matrix

Example 1: Find the value of Q − 2P, if [Tex]P = \left[\begin{array}{c} 5\\ 7\\ -9 \end{array}\right]_{3\times1}and Q = \left[\begin{array}{c} 11\\ 21\\ 31 \end{array}\right]_{3\times1} [/Tex]

Solution:

[Tex]Q − 2P = \left[\begin{array}{c} 11\\ 21\\ 31 \end{array}\right]_{3\times1} − 2 \times\left[\begin{array}{c} 5\\ 7\\ -9 \end{array}\right]_{3\times1}[/Tex]

[Tex]Q − 2P = \left[\begin{array}{c} 11\\ 21\\ 31 \end{array}\right]_{3\times1}− \left[\begin{array}{c} 10\\ 14\\ -18 \end{array}\right]_{3\times1}[/Tex]

[Tex]Q − 2P = \left[\begin{array}{c} (11-10)\\ (21-14)\\ (31-(-18)) \end{array}\right]_{3\times1}[/Tex]

[Tex]Q − 2P = \left[\begin{array}{c} 1\\ 7\\ 49 \end{array}\right]_{3\times1}[/Tex]

Example 2: Prove that the transpose of a column matrix is a row matrix.

Solution:

Let us consider an example, to prove that the transpose of a column matrix is a row matrix.

[Tex]A = \left[\begin{array}{c} 15\\ 0\\ -13 \end{array}\right]_{3\times1}[/Tex]

The above matrix is a column matrix of order “3 × 1.” We know that the transpose of a matrix is obtained by interchanging the entries of rows and columns. So, the order of the transpose of the given matrix will be “1 × 3.”

[Tex]A = \left[\begin{array}{c} 15\\ 0\\ -13 \end{array}\right]_{3\times1}⇒ A^{T} = \left[\begin{array}{ccc} 15 & 0 & -13\end{array}\right]_{1\times3}[/Tex]

We can see that the resultant matrix is a row matrix.

Hence proved.

Example 3: Find the product of the matrices given below.

[Tex]A = \left[\begin{array}{c} 4\\ -5\\ 6 \end{array}\right]_{3\times1} and B = \left[\begin{array}{ccc} 1 & 0 & 2\end{array}\right]_{1\times3}[/Tex]

Solution:

[Tex]A × B = \left[\begin{array}{c} 4\\ -5\\ 6 \end{array}\right]_{3\times1}\times\left[\begin{array}{ccc} 1 & 0 & 2\end{array}\right]_{1\times3}[/Tex]

[Tex]A × B = \left[\begin{array}{ccc} 4\times1 & 4\times0 & 4\times2\\ -5\times1 & -5\times0 & -5\times2\\ 6\times1 & 6\times0 & 6\times2 \end{array}\right]_{3\times3}[/Tex]

[Tex]A × B = \left[\begin{array}{ccc} 4 & 0 & 8\\ -5 & 0 & -10\\ 6 & 0 & 12 \end{array}\right]_{3\times3}[/Tex]

Example 4: Find the value of M − N, if 

[Tex]M = \left[\begin{array}{c} 81\\ 72\\ 63 \end{array}\right]_{3\times1} and N = \left[\begin{array}{c} -48\\ 36\\ 21 \end{array}\right]_{3\times1}[/Tex]

Solution:

[Tex]M – N = \left[\begin{array}{c} 81\\ 72\\ 63 \end{array}\right]_{3\times1}- \left[\begin{array}{c} -48\\ 36\\ 21 \end{array}\right]_{3\times1}[/Tex]

[Tex]M – N = \left[\begin{array}{c} 81+48\\ 72-36\\ 63-21 \end{array}\right]_{3\times1}[/Tex]

[Tex]M – N = \left[\begin{array}{c} 129\\ 36\\ 42 \end{array}\right]_{3\times1}[/Tex]

A rectangular array of numbers that are arranged in rows and columns is known as a “matrix.” The size of a matrix can be determined by the number of rows and columns in it. If a matrix has “m” rows and “n” columns, then it is said to be an “m by n” matrix and is written as an “m × n” matrix.

For example, if a matrix has five rows and three columns, it is a “5 × 3” matrix. We have various types of matrices, like rectangular, square, triangular, symmetric, singular, etc. Now let us discuss the column matrix in detail.