Matrices – Complete Revision Series

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Matrices – Complete Revision Series (JEE Level)

Matrices provide a powerful algebraic language to represent systems of equations, transformations, and logical structure. In JEE, this chapter blends precision with speed.

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1. Matrix Definition

\[ A = [a_{ij}]_{m \times n} \]

A matrix is a rectangular array of elements arranged in rows and columns.

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2. Types of Matrices

• Row Matrix: \(1 \times n\)
• Column Matrix: \(m \times 1\)
• Square Matrix: \(n \times n\)
• Zero Matrix
• Diagonal Matrix
• Scalar Matrix
• Identity Matrix \(I_n\)
• Symmetric Matrix: \(A^T = A\)
• Skew-Symmetric Matrix: \(A^T = -A\)

Important: Diagonal elements of a skew-symmetric matrix are always zero.

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3. Matrix Operations

Addition

\[ A+B = [a_{ij}+b_{ij}] \]

Scalar Multiplication

\[ kA = [ka_{ij}] \]

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4. Matrix Multiplication

If \(A_{m\times n}\) and \(B_{n\times p}\), then

\[ (AB)_{ij} = \sum_{k=1}^{n} a_{ik}b_{kj} \]

• Not commutative: \(AB \neq BA\)
• Associative
• Distributive

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5. Transpose

\[ (A^T)_{ij} = a_{ji} \]

• \((A^T)^T = A\)
• \((AB)^T = B^T A^T\)
• \((A+B)^T = A^T + B^T\)

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6. Adjoint of a Matrix

The adjoint (adjugate) of a square matrix \(A\), denoted by \(\text{Adj}\,A\), is the transpose of its cofactor matrix.

Definition

\[ (\text{Adj}\,A)_{ij} = C_{ji} \] where \[ C_{ij} = (-1)^{i+j} M_{ij} \] and \(M_{ij}\) is the minor of element \(a_{ij}\).

Key Property

\[ A(\text{Adj}\,A) = (\text{Adj}\,A)A = |A| I \]

JEE Gold Formula: This identity is the backbone of inverse and proof questions.

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7. Important Adjoint Properties

• \(\text{Adj}(AB) = (\text{Adj}B)(\text{Adj}A)\)
• \(\text{Adj}(A^T) = (\text{Adj}A)^T\)
• \(\text{Adj}(kA) = k^{n-1}\text{Adj}(A)\) for \(n \times n\) matrix
• \(|\text{Adj}(A)| = |A|^{n-1}\)
• \(\text{Adj}(I) = I\)

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8. Inverse of a Matrix

A square matrix \(A\) is invertible if

\[ |A| \neq 0 \]

Formula

\[ A^{-1} = \frac{1}{|A|}\text{Adj}(A) \]

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9. Solving Linear Equations

\[ AX = B \Rightarrow X = A^{-1}B \quad (|A|\neq0) \]

Link with Determinants: Zero determinant → no inverse → system fails or infinite solutions.

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10. High-Yield JEE Formulas

• \(A(\text{Adj}A)=|A|I\)
• \((A^{-1})^{-1}=A\)
• \((AB)^{-1}=B^{-1}A^{-1}\)
• \((A^T)^{-1}=(A^{-1})^T\)
• \(|A^{-1}|=\frac{1}{|A|}\)

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11. Common Mistakes

• Forgetting determinant condition
• Wrong cofactor signs
• Assuming commutativity
• Mixing adjoint and transpose

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