Revision: 12th Std >> Matrices MAH-MHT CET (PCM/PCB) Matrices

If A and B are non-singular square matrices of the same order such that AB = BA = I (where I is the identity matrix of the same order as A and B), then A and B are called inverses of each other.

We write A⁻¹ = B and B⁻¹ = A.

i.e. AA⁻¹ = A⁻¹A = I.

• If |A| ≠ 0, then A⁻¹ exists.
• If the inverse of a square matrix exists, then it is unique. A matrix can not have more than one distinct inverse.

Let A = [aij] be a square matrix of order n. Then, the minor M_ij of a_ij in A is the determinant obtained by deleting the i^th row and the j^th column in which element a_ij lies. It is denoted by M_ij​ of A.

Let A = [a_ij] be a square matrix of order n. Then, the cofactor C_ij (or A_ij) of a_ij in A is (−1)^i+j times M_ij, where M_ij is the minor of a_ij in A.

∴ C_ij = (−1)^i+j M_ij

The adjoint of A is defined as the transpose (i.e. interchange rows and columns) of the cofactor matrix, and it is denoted by adj (A).

The transpose of a matrix is obtained by interchanging its rows and columns.

• If a matrix is A, its transpose is denoted by A^T

• If A is of order m × n, then
  A^T is of order n × m

• First row of A becomes first column of A^T, and so on.

Skew-Symmetric Matrix: A square matrix A = [aij] n×n is skew-symmetric if Aᵀ = −A i.e., aij = −aji for all i and j.

A square matrix A = [aᵢⱼ]ₙ×ₙ is symmetric if Aᵀ = A
i.e., aᵢⱼ = aⱼᵢ for all i and j.

By Adjoint Method: \[A^{-1}=\frac{\mathrm{adj}A}{|A|}\]

By Using Algebraic Equation: A matrix A and an algebraic equation in matrix A is in the form of A² + bA + C = O.

\[A^{-1}=\frac{1}{C}\left[-aA-bI\right]\]

\[A= \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}\]

\[|A|=a_{11}(a_{22}a_{33}-a_{32}a_{23})-a_{12}(a_{21}a_{33}-a_{31}a_{23})+a_{13}(a_{21}a_{32}-a_{31}a_{22})\]

• If |A| = 0
  A matrix is called a Singular Matrix
• If |A| ≠ 0
  Matrix is called a Non-Singular Matrix

1. adj (AB) = (adj B) (adj A)
2. (adj A)A = A (adj A) = |A| Iₙ
3. (a) |adj A| = |A|ⁿ⁻¹, if |A| ≠ 0
   (b) |adj A| = 0, if |A| = 0
4. If |A| = 0, then (adj A) A = A (adj A) = O
5. adj (Aᵐ) = (adj A)ᵐ, m ∈ N
6. adj (kA) = kⁿ⁻¹ (adj A), k ∈ R
7. adj (adj A) = |A|ⁿ⁻² A, A is non-singular matrix
8. adj (adj A) = |A|ⁿ⁻² A, A is non-singular matrix

Method of Inversion

• Given: AX = B
• Multiply both sides by A⁻¹

Result:

• X = A⁻¹B

Key Steps:

• Pre-multiply by A⁻¹
• Use property: A⁻¹A = I
• Final solution: X = A⁻¹B

Method of Reduction

• No need to find A⁻¹
• Apply elementary row operations to the matrix. Process:

• Convert the matrix into upper triangular form
• System reduces to:
  • b₁₁x + b₁₂y + b₁₃z = b₁′
  • b₂₂y + b₂₃z = b₂′
  • b₃₃z = b₃′

Final Step:

• Solve using back substitution:
  • First find z
  • Then y
  • Then x