Determinants and Matrices — MHT-CET Maths

Formulas (5)

• Determinants and Cofactors: Expansion Along a Row · Expansion and cofactors
  $$|A| = \sum_{j} a_{ij}A_{ij}, \qquad A_{ij} = (-1)^{i+j}M_{ij}, \qquad \sum_j a_{ij}A_{kj} = 0 \ (k \ne i)$$
• The Adjoint and A·adj(A) = |A|·I · The adjoint identity
  $$A\,\operatorname{adj}A = |A|\,I \qquad \operatorname{adj}\begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \qquad (\operatorname{adj}A)^{-1} = \frac{A}{|A|}$$
• |adj A| = |A|ⁿ⁻¹ and |kA| = kⁿ|A| · Determinant identities
  $$|\operatorname{adj}A| = |A|^{n-1} \qquad |kA| = k^n|A| \qquad |AB| = |A||B| \qquad |A^{-1}| = \frac{1}{|A|}$$
• A·adj(A) = AAᵀ: Two Equations From the Diagonal and Off-Diagonal · The AAᵀ condition
  $$A\,\operatorname{adj}A = AA^T \iff AA^T = |A|\,I \iff (\text{off-diagonal of } AA^T) = 0 \ \text{and}\ (\text{diagonal of } AA^T) = |A|$$
• Determinant Equations: When Does |A| Vanish? · Vanishing determinant
  $$|A| = 0 \iff A \text{ singular} \iff A^{-1} \text{ does not exist} \qquad 1 + \omega + \omega^2 = 0,\ \omega^3 = 1$$

Watch out for (5)

• Reading (adj A)₂₃ as the cofactor A₂₃
  → Determinants and Cofactors: Expansion Along a Row
• Substituting the relations before expanding
  → The Adjoint and A·adj(A) = |A|·I
• Using |adj A| = |A|
  → |adj A| = |A|ⁿ⁻¹ and |kA| = kⁿ|A|
• Equating AAᵀ to |A| only on the diagonal
  → A·adj(A) = AAᵀ: Two Equations From the Diagonal and Off-Diagonal
• Stopping at cos 2B = 0
  → Determinant Equations: When Does |A| Vanish?

Formulas (5)

• Inverse of a 2 × 2 Matrix · 2 × 2 inverse
  $$\begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} = \frac{1}{ad - bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}, \quad ad - bc \ne 0$$
• Invert an Expression: Compute A² − 5A or A + B First, Then Invert · Order of operations
  $$(A^2 - 5A)^{-1} = \left(\text{the matrix } A^2 - 5A\right)^{-1} \ne (A^2)^{-1} - (5A)^{-1}, \qquad (A + B)^{-1} \ne A^{-1} + B^{-1}$$
• Matrices with tan x Entries: |A| = sec²x and adj A = Aᵀ · The tan-entry matrix
  $$A = \begin{pmatrix} 1 & \tan x \\ -\tan x & 1 \end{pmatrix}:\ |A| = \sec^2x,\ \operatorname{adj}A = A^T,\ A^{-1} = \cos^2x\,A^T,\ A^TA^{-1} = \begin{pmatrix} \cos 2x & -\sin 2x \\ \sin 2x & \cos 2x \end{pmatrix}$$
• (AB)⁻¹ = B⁻¹A⁻¹: Inverting Products and Recovering a Factor · Inverse of a product
  $$(AB)^{-1} = B^{-1}A^{-1} \qquad B^{-1} = (AB)^{-1}A \qquad (A^{-1})^{-1} = A \qquad (A^T)^{-1} = (A^{-1})^T$$
• Unknown Entries and A⁻¹ = A³: Use AA⁻¹ = I · The defining property of the inverse
  $$AB = I \iff B = A^{-1} \qquad A^{-1} = A^{3} \iff A^{4} = I$$

Watch out for (5)

• Swapping signs on the diagonal instead of the off-diagonal
  → Inverse of a 2 × 2 Matrix
• Distributing the inverse over a sum
  → Invert an Expression: Compute A² − 5A or A + B First, Then Invert
• Sign of the sin 2x entries
  → Matrices with tan x Entries: |A| = sec²x and adj A = Aᵀ
• Keeping the order
  → (AB)⁻¹ = B⁻¹A⁻¹: Inverting Products and Recovering a Factor
• Solving all nine entries
  → Unknown Entries and A⁻¹ = A³: Use AA⁻¹ = I

Formulas (4)

• Cayley–Hamilton for 2 × 2: A² − (tr A)A + |A|·I = 0 · Cayley–Hamilton (2 × 2)
  $$A^2 - (\operatorname{tr}A)\,A + |A|\,I = O \qquad A^{-1} = \frac{(\operatorname{tr}A)\,I - A}{|A|}$$
• A⁻¹ = αI + βA: Read α and β From the Theorem · Inverse as a combination
  $$A^{-1} = \alpha I + \beta A \ \text{ with } \ \alpha = \frac{\operatorname{tr}A}{|A|},\ \beta = -\frac{1}{|A|}$$
• A Factored Polynomial in A Gives A⁻¹ in One Line · From a polynomial relation to the inverse
  $$A^2 - pA + qI = O \ (q \ne 0) \ \Rightarrow\ A^{-1} = \frac{pI - A}{q} \qquad (A + kI)^{-1}:\ \text{write } A^2 - pA + qI = (A + kI)(A - mI) + cI$$
• Powers of a Matrix: Find the Cycle · Powers modulo a cycle
  $$A^m = cI \ \Rightarrow\ A^{mk + r} = c^k A^r \qquad \begin{pmatrix} i & 1 \\ 1 & 0 \end{pmatrix}^3 = iI,\quad i^4 = 1$$

Watch out for (4)

• Sign of the determinant term
  → Cayley–Hamilton for 2 × 2: A² − (tr A)A + |A|·I = 0
• α from the wrong entry
  → A⁻¹ = αI + βA: Read α and β From the Theorem
• Concluding A = 3I or A = 5I
  → A Factored Polynomial in A Gives A⁻¹ in One Line
• Reducing 2029 modulo 3 instead of 12
  → Powers of a Matrix: Find the Cycle

Formulas (3)

• Solving AX = B: Elimination, or X = A⁻¹B · Linear system in matrix form
  $$AX = B,\quad |A| \ne 0 \ \Rightarrow\ X = A^{-1}B \ \text{(unique)}$$
• Homogeneous Systems: Non-Trivial Solutions Need |A| = 0 · Homogeneous system
  $$AX = O \text{ has a non-trivial solution} \iff |A| = 0$$
• Symmetric + Skew-Symmetric: The Unique Split, and Why Odd-Order Skew Is Singular · Symmetric and skew-symmetric parts
  $$M = \frac{M + M^T}{2} + \frac{M - M^T}{2} \qquad B^T = -B,\ n \text{ odd} \ \Rightarrow\ |B| = 0$$

Watch out for (3)

• Trusting a solution without checking every equation
  → Solving AX = B: Elimination, or X = A⁻¹B
• Reading |A| = 0 as 'no solution'
  → Homogeneous Systems: Non-Trivial Solutions Need |A| = 0
• Expecting a 2 × 2 skew-symmetric matrix to be singular
  → Symmetric + Skew-Symmetric: The Unique Split, and Why Odd-Order Skew Is Singular