উদাহরণ: $2\times n$ ম্যাট্রিক্সকে $n\times 2$ ম্যাট্রিক্স দিয়ে গুণ করা

What is a Zero Matrix?

A zero matrix, or null matrix, is an $m \times n$ matrix where all the entries are zero. A zero matrix can be any size, unlike identity matrices, which are only square matrices. For example, all of the following are zero matrices: $$\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$$ $$\begin{pmatrix} 0 & 0 & 0\\ 0 & 0 & 0 \end{pmatrix}$$ $$\begin{pmatrix} 0 \\ 0 \end{pmatrix}$$ $$\begin{pmatrix} 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{pmatrix}$$

That's a whole lot of nothing.

For any matrix $A_{mn}$, if we add $0_{mn}$ to it, we just get $A_{mn}$! You can liken this to adding $0$ to a number to get the same number!

You will need this in the future. When you are learning about vectors in linear algebra, you may want to make a transformation where you send all vectors to the zero vector. In that case, you would use a zero matrix! But for now, you don't have to worry about linear algebra.

(Video) What is a Zero Matrix?

by Wrath of Math

This video by Wrath of Math explains what a zero matrix is and how to use it.

Summary

A zero matrix is a matrix whose entries are all equal to $0$. We can also call this a null matrix.

Here is example of a zero matrix:

$$\begin{bmatrix} 0&0&0 \\ 0&0&0 \\ \end{bmatrix}$$

Here is another example of a zero matrix:

$$\begin{bmatrix} 0 \\ 0 \\ 0 \\ \end{bmatrix}$$

If we go back to what we learned a few years ago, you'll remember that anything multiplied by $0$ is $0$. The same goes with zero matrices!

$$\begin{bmatrix} 1&2 \\ 3&4 \\ 5&6 \\ \end{bmatrix} \cdot \begin{bmatrix} 0&0 \\ 0&0 \\ \end{bmatrix} = \begin{bmatrix} 0&0 \\ 0&0 \\ 0&0 \\ \end{bmatrix}$$

Any time we multiply a given matrix by a zero matrix, you will always get a zero matrix.

Let's look at what happens when we add a zero matrix to a given matrix. We know with regular addition, we use the Identity Property of Addition, where we get the number itself. For example, $3+0=3$.

$$\begin{bmatrix} 1&2 \\ 3&4 \\ \end{bmatrix} + \begin{bmatrix} 0&0 \\ 0&0 \\ \end{bmatrix} = \begin{bmatrix} 1&2 \\ 3&4 \\ \end{bmatrix}$$

The same goes for when subtracting a $0$ matrix!

Zero Matrices

Zero matrices are matrices with all entries being $0$.

The following are examples of zero matrices:

$$\begin{bmatrix} 0&0&0 \\ 0&0&0 \\ \end{bmatrix}$$ $$\begin{bmatrix} 0&0&0&0&0 \\ \end{bmatrix}$$ $$\begin{bmatrix} 0&0 \\ 0&0 \\ 0&0 \\ 0&0 \\ \end{bmatrix}$$

The size of the matrix doesn't matter. We can have square matrices, long matrices, and short matrices.