Q.If $A$ is a matrix of order $3 \times 3$, then the number of minors in the determinant of $A$ are ________ .

The key idea is that a minor is the determinant of a submatrix obtained by deleting one row and one column. For a $3 \times 3$ matrix, each entry has exactly one minor associated with it.

Step 1: A $3 \times 3$ matrix has $3 \times 3 = 9$ entries. …

A minor in a determinant is the determinant of a smaller square matrix obtained by deleting exactly one row and one column from the original matrix. For a $3 \times 3$ matrix, each minor corresponds to a specific pair: the row you remove and the column you remove.

1. Understand what a minor is.

   If $A = [a_{ij}]$ is a $3 \times 3$ matrix, the minor $M_{ij}$ is the determinant of the $2 \times 2$ matrix left after deleting the $i$-th row and $j$-th column. So each minor is tied to a specific position $(i, j)$.

2. Count the possible $(i, j)$ pairs.

   The row index $i$ can be 1, 2, or 3 — that’s 3 choices.

   The column index $j$ can also be 1, 2, or 3 — another 3 choices.

   So the total number of distinct minors is $3 \times 3 = 9$.

3. Check with a concrete example. …

Method: Counting Minors of a Square Determinant by Position, Not by Expansion Path

To count how many minors an $n\times n$ determinant has in total, count the number of positions (row, column) that can be deleted — not the number of terms that happen to appear in any one particular cofactor expansion.

Steps

Step 1: Recall what a minor is tied to

The minor $M_{ij}$ is defined for each entry $a_{ij}$ of the matrix: it's the determinant left after deleting row $i$ and column $j$. Every entry of the matrix has its own minor, whether or not that entry ever appears in a particular expansion you choose to write out.

Step 2: Count the number of independent choices for $(i,j)$ …

Common Mistakes

Mistake 1: Counting only the minors used in one cofactor expansion (e.g. just the first row)

Why it's wrong: when computing a determinant by hand, you typically only ever write out the 3 minors for one row (say $M_{11}, M_{12}, M_{13}$) — this can leave the false impression that a $3\times3$ matrix "has" only 3 minors. In reality every one of the 9 entries has its own minor, regardless of which row or column you happen to expand along to compute the determinant's value. Correct approach: separate "how many minors exist" (always $n^2$, tied to every entry) from "how many minors you actually use" when computing one particular expansion (only $n$, tied to one row or column).

Mistake 2: Confusing "minors" with "cofactors" and expecting a different count …