Matrices & Determinants

Linear Algebraic Viewpoint

A matrix represents a linear transformation between vector spaces. The determinant $det(A)$ quantifies the oriented scaling factor of volume under this transformation. In competitive exams, mastery spans determinant row/column operations, adjoint-determinant identities, characteristic polynomials, and solvability of linear systems.

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1. Properties of Determinants

📊Visual Mathematical Intuition

Determinant as Signed Parallelogram Area & Linear Mapping

Signed Area:$\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc = \text{Area}(\vec{u}, \vec{v})$

Orientation:$\det(A) > 0 \implies \text{Preserves}, \; \det(A) = 0 \implies \text{Collapses to Line}$

A $2 \times 2$ determinant equals the signed area of the parallelogram spanned by column vectors $\vec{u}$ and $\vec{v}$. When $\det(A) = 0$, the parallelogram collapses into a degenerate 1D line.

1. Transpose Invariance: $det(A^T)=det(A)$
2. Row/Column Interchange: Swapping two rows/columns flips the sign of $det(A)$.
3. Scalar Multiplication: $det(kA)=k^{n}det(A)$ for an $n\times n$ matrix $A$.
4. Multiplicative Property: $det(AB)=det(A)\cdot det(B)$
5. Elementary Operations: Adding a scalar multiple of one row to another preserves $det(A)$.

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2. Adjoint and Inverse of a Matrix

For an $n\times n$ invertible matrix $A$ ($det(A)\ne 0$):

$$A\cdot \text{adj}(A)=\text{adj}(A)\cdot A=det(A)\cdot I_n$$$$A^{-1}=\frac{1}{det(A)}\text{adj}(A)$$

Critical Adjoint Identities

Identity	Formula
Determinant of Adjoint	$det(\text{adj}(A))=(det(A){)}^{n-1}$
Adjoint of Adjoint	$\text{adj}(\text{adj}(A))=(det(A){)}^{n-2}A$
Determinant of Double Adjoint	$det(\text{adj}(\text{adj}(A)))=(det(A){)}^{(n-1{)}^2}$
Inverse of Transpose	$(A^T{)}^{-1}=(A^{-1}{)}^T$
Reversal Law	$(AB{)}^{-1}=B^{-1}{A}^{-1}$ and $\text{adj}(AB)=\text{adj}(B)\cdot \text{adj}(A)$

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3. Cayley-Hamilton Theorem & Characteristic Polynomial

Cayley-Hamilton Theorem

Every square matrix $A$ satisfies its own characteristic equation:

$$det(\lambda I-A)=0⟹{\lambda }^n+c_{n-1}{\lambda }^{n-1}+\cdots +c_0=0⟹A^n+c_{n-1}{A}^{n-1}+\cdots +c_{0}I=0$$

For a $2\times 2$ matrix $A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$:

$$A^2-\text{tr}(A)A+det(A)I=0\text{where }\text{tr}(A)=a+d$$

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4. System of Linear Equations (Solvability Criteria)

Consider $AX=B$ with coefficient matrix $A$ and augmented matrix $[A|B]$:

1. Unique Solution: $det(A)\ne 0$ (Rank $A$ = Rank $[A|B]$ = $n$).
2. Infinitely Many Solutions (Consistent): $det(A)=0$ AND $(\text{adj}(A))B=\mathbf{0}$ (Rank $A$ = Rank $[A|B]<n$).
3. No Solution (Inconsistent): $det(A)=0$ AND $(\text{adj}(A))B\ne \mathbf{0}$ (Rank $A<\text{Rank }[A|B]$).