Null Matrix

If each element of an m × n matrix be 0, the null element of F, the matrix is said to be the null matrix or the zero matrix of order m × n and it is denoted by Om,n. It is also denoted by O, when no confusion regarding its order arises.

Null or zero Matrix: Whether A is a rectangular or square matrix, A - A is a matrix whose every element is zero. The matrix whose every element is zero is called a null or zero matrix and it is denoted by 0.

Thus for A and 0 of the same order we have A + 0 = A

For example,

\(\begin{bmatrix} 0 & 0 \end{bmatrix}\) is a zero matrix of order 1 × 2.

\(\begin{bmatrix} 0\\ 0 \end{bmatrix}\) is a zero or null matrix of order 2 × 1.

\(\begin{bmatrix} 0 & 0\\ 0 & 0 \end{bmatrix}\) is a null matrix of order 2 × 2.

\(\begin{bmatrix} 5 & 6 & 4\\ 1 & 0 & 9 \end{bmatrix}\) is a null matrix of order 2 × 3.


Problems on Null or zero matrix:

1. Find two nonzero matrices whose product is a zero matrix.

Solution:

Let A = \(\begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}\) and B = \(\begin{bmatrix} 0 & 0\\ 0 & 1 \end{bmatrix}\) be two non-zero matrices.

But AB = \(\begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}\) \(\begin{bmatrix} 0 & 0\\ 0 & 1 \end{bmatrix}\) = \(\begin{bmatrix} 0 & 0\\ 0 & 0 \end{bmatrix}\) is a null matrix.

 

2. If A = \(\begin{bmatrix} 1 & 2\\ -1 & -1 \end{bmatrix}\), show that A2 + I = 0.

(I and 0 being identity and null matrices of order 2).

Solution:

Given, A = \(\begin{bmatrix} 1 & 2\\ -1 & -1 \end{bmatrix}\)

Now A2 = \(\begin{bmatrix} 1 & 2\\ -1 & -1 \end{bmatrix}\)\(\begin{bmatrix} 1 & 2\\ -1 & -1 \end{bmatrix}\) = \(\begin{bmatrix} -1 & 0\\ 0 & -1 \end{bmatrix}\)

Therefore, A2 + I = \(\begin{bmatrix} -1 & 0\\ 0 & -1 \end{bmatrix}\) + \(\begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}\) = \(\begin{bmatrix} 0 & 0\\ 0 & 0 \end{bmatrix}\)

Thus, A2 + I = 0.





10th Grade Math

From Null Matrix to HOME PAGE


New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.



Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



Share this page: What’s this?

Recent Articles

  1. Measuring Capacity | Standard Unit of Capacity | Litre | Millilitres

    Nov 29, 23 01:15 AM

    2 Tablespoonful of Water
    We will discuss about measuring capacity. The milkman measures milk in liters. Petrol is given in liters. Mobil oil is sold in liters. Two milk bottles contain 1 liter of milk. One milk bottle

    Read More

  2. Addition and Subtraction of Units of Measurement | Metric Units

    Nov 29, 23 12:54 AM

    Addition of Lengths
    We can add the units of measurement like decimal numbers. 1. Add 5 m 9 dm and 11 m and 5 dm Solution: 5 m 9 dm = 5.9 m 11 m 5 dm = 11.5 m Hence, 5 m 9 dm + 11 m 5 dm = 17 m 4 dm or 17.4 m 2. Add 15 cm…

    Read More

  3. 1 to 10 Times Tables | 1 - 10 Times Table Chart |Multiplication Tables

    Nov 29, 23 12:50 AM

    1 to 10 Times Tables
    Memorizing 1 to 10 Times Tables are very important for mental math and quick calculations. Times Tables are used during multiplication and division. Let us learn all the times tables from 1 to 10 to i…

    Read More