Determinant | Lesson Note

1. Principal & Secondary Diagonals

Before defining a determinant, let us understand the principal (leading) and secondary diagonals of a matrix.

2. Activity — Discovering the Determinant

Given:

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

Observation:

Product of elements in principal diagonal
$d_1 = 1 \times 3 = 3$

Product of elements in secondary diagonal
$d_2 = 4 \times 2 = 8$

$$d_1 - d_2 = 3 - 8 = -5$$

Conclusion

$-5$ is a scalar value, which is popularly known as the Determinant of matrix $A$.

3. Definition of Determinant

The scalar value obtained by subtracting the product of elements in the secondary diagonal from the product of elements in the principal diagonal is called the Determinant of a matrix.

Let $A$ be any square matrix. Its determinant is written as $|A|$ or $\det A$.

$$A = \begin{bmatrix}a & b\\ c & d\end{bmatrix}$$

Formula for 2×2 Determinant

$$|A| = \begin{vmatrix}a & b\\ c & d\end{vmatrix} = ad - bc$$

💡 Determinants are useful for solving systems of linear equations.

4. Determinant of a 1×1 Matrix

For a 1×1 matrix $A = [a]$, the determinant equals the element itself:

$$A = [a] \;\Rightarrow\; |A| = |a| = a$$

Examples:

$A = [a] \;\Rightarrow\; |A| = a$
$B = [-2] \;\Rightarrow\; |B| = |-2| = -2$
$C = [5] \;\Rightarrow\; |C| = |5| = 5$

Note on Absolute Value vs. Determinant

The absolute value of a number is always non-negative:

$$|a| = a \quad \text{and} \quad |-a| = a$$

⚠️ Important: Do not confuse the determinant notation $|A|$ with absolute value. For matrices, $|A|$ means the determinant — which can be negative. Absolute value is always positive.

5. Activity — Singular & Non-Singular Matrices

Given:

$$A = \begin{bmatrix}2 & 3\\ 2 & 3\end{bmatrix}, \quad B = \begin{bmatrix}4 & 5\\ 6 & 7\end{bmatrix}$$

Observation:

$$|A| = \begin{vmatrix}2 & 3\\ 2 & 3\end{vmatrix} = 2\times3 - 2\times3 = 6 - 6 = 0$$

$$|B| = \begin{vmatrix}4 & 5\\ 6 & 7\end{vmatrix} = 4\times7 - 6\times5 = 28 - 30 = -2$$

Conclusion

$|A| = 0$ ⟹ Matrix $A$ is a Singular Matrix.
$|B| = -2 \neq 0$ ⟹ Matrix $B$ is a Non-Singular Matrix.

6. Singular & Non-Singular Matrices

Singular Matrix

A matrix whose determinant is zero is called a Singular Matrix.

$$|A| = 0$$

Example: $A = \begin{bmatrix}2&3\\2&3\end{bmatrix}$, $|A|=0$

Non-Singular Matrix

A matrix whose determinant is non-zero is called a Non-Singular Matrix.

$$|A| \neq 0$$

Example: $B = \begin{bmatrix}4&5\\6&7\end{bmatrix}$, $|B|=-2$

7. Summary

Term	Meaning	Key Formula / Condition
Principal diagonal	Elements from top-left to bottom-right	$a_{11},\, a_{22},\, a_{33},\ldots$
Secondary diagonal	Elements from top-right to bottom-left	$a_{1n},\, a_{2,n-1},\ldots$
Determinant	Scalar value from a square matrix	$\|A\| = ad - bc$
Notation	Two standard ways to write it	$\|A\|$ or $\det A$
Singular matrix	Determinant equals zero	$\|A\| = 0$
Non-singular matrix	Determinant is non-zero	$\|A\| \neq 0$
Application	Solving linear equations	Cramer's Rule, etc.