Q.How many distinct $2\times 2$ matrices can be formed by using numbers 5, 7 and $-1$? Justify your answer.

Counting Matrix Entries – A First Look

A matrix is just a rectangular array of numbers. If you have a matrix with $m$ rows and $n$ columns, the total number of entries is simply $m \times n$.

That is the whole idea. But why does this matter, and what subtlety hides behind it?

The Intuition

Imagine a classroom with rows of desks. If there are 5 rows and each row has 4 desks, you know there are $5 \times 4 = 20$ desks in total. A matrix works exactly the same way: each row has exactly as many entries as there are columns, and every column has exactly as many entries as there are rows.

So for a $3 \times 2$ matrix:

there are 3 rows and 2 columns, giving $3 \times 2 = 6$ entries. You can count them: a, b, c, d, e, f.

The Precise Statement

That's it. There is no trick. Every matrix of size $m \times n$ has exactly $mn$ entries.

Why This Matters

You will use this idea constantly:

• Matrix multiplication: When multiplying $A$ ($m \times n$) by $B$ ($n \times p$), the result has $m \times p$ entries. Each entry is a sum of $n$ products.
• Determinants: Only square matrices ($n \times n$) have determinants. A $2 \times 2$ matrix has 4 entries, a $3 \times 3$ has 9, and so on. …