Determinant | Lesson Note

Determinants | Additional Maths · Class 10 | New Millennium Academy

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📐 Additional Maths · Class 10

📌 1. Principal & Secondary Diagonals

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

2×2 Example

$$ \begin{bmatrix} \color{#e6b422}{a_{11}} & a_{12} \\ a_{21} & \color{#e6b422}{a_{22}} \end{bmatrix} $$

Principal diagonal: $a_{11}, a_{22}$

3×3 Example

$$ \begin{bmatrix} \color{#e6b422}{1} & 2 & 3 \\ 4 & \color{#e6b422}{5} & 6 \\ 7 & 8 & \color{#e6b422}{9} \end{bmatrix} $$

Principal: 1,5,9 | Secondary: 3,5,7

Secondary diagonal

$$ \begin{bmatrix} 1 & 2 & \color{#e6b422}{3} \\ 4 & \color{#e6b422}{5} & 6 \\ \color{#e6b422}{7} & 8 & 9 \end{bmatrix} $$

🧪 2. Activity — Discovering the Determinant

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

📐 Product of principal diagonal $ d_1 = 1 \times 3 = 3 $

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

✨ $ d_1 - d_2 = 3 - 8 = \mathbf{-5} $

✅ Conclusion: $-5$ is a scalar value, 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 square 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 (Cramer's Rule, invertibility, etc.).

🔢 4. Determinant of a 1×1 Matrix

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

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

Example: $A = [5]$ → $|A| = 5$

Example: $B = [-2]$ → $|B| = -2$

Example: $C = [0]$ → $|C| = 0$

⚠️ 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 non-negative.

Example: $|[-3]| = -3$ (determinant), while absolute value $|-3| = 3$.

⚡ 5. Activity — Singular & Non-Singular Matrices

Given matrices:

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

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

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

✅ Conclusion: $|A| = 0$ ⟹ Matrix $A$ is Singular. $|B| = -2 \neq 0$ ⟹ Matrix $B$ is Non‑Singular.

📌 6. Singular & Non-Singular Matrices

🔴 Singular Matrix

A matrix whose determinant is zero.
$|A| = 0$

Example: $\begin{bmatrix}2&3\\2&3\end{bmatrix}$

🟡 Non‑Singular Matrix

A matrix whose determinant is non-zero.
$|A| \neq 0$

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

📋 7. Summary – Key Concepts

Term	Meaning	Key Formula / Condition
Principal diagonal	Top-left to bottom-right	$a_{11}, a_{22}, a_{33}, \dots$
Secondary diagonal	Top-right to bottom-left	$a_{1n}, a_{2,n-1}, \dots$
Determinant	Scalar value from a square matrix	$\|A\| = ad - bc$ (2×2)
Notation	Two standard ways	$\|A\|$ or $\det A$
Singular matrix	Determinant equals zero	$\|A\| = 0$
Non‑singular matrix	Determinant is non‑zero	$\|A\| \neq 0$
Application	Solving linear systems	Cramer's Rule, inverse existence

🌟 Did you know? Determinants extend to 3×3 and larger matrices using expansion (Laplace). They reveal invertibility, area/volume scaling, and much more — a gateway to linear algebra!

Term	Meaning	Key Formula / Condition
Principal diagonal	Top-left to bottom-right	\(a_{11}, a_{22}, a_{33}, \dots\)
Secondary diagonal	Top-right to bottom-left	\(a_{1n}, a_{2,n-1}, \dots\)
Determinant	Scalar value from a square matrix	\(\|A\| = ad - bc\) (2×2)
Notation	Two standard ways	\(\|A\|\) or \(\det A\)
Singular matrix	Determinant equals zero	\(\|A\| = 0\)
Non‑singular matrix	Determinant is non‑zero	\(\|A\| \neq 0\)
Application	Solving linear systems	Cramer's Rule, inverse existence