Revision: Matrices Maths and Stats HSC Science (General) 12th Standard Board Exam Maharashtra State Board

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).

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.

Two matrices are equivalent if one can be obtained from the other by a finite number of elementary operations

• Denoted by: A ∼ B

If A = [aij], then the negative of A, denoted by −A, is the matrix obtained by replacing each element a_ij by −a_ij

−A = [−a_ij]

• Order of −A = order of A

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.

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

Minor

Delete ith row and jth column: M_ij​

Cofactor of a_ij

A_ij = (−1)^i+j × (minor of a_ij)

Sign pattern:

\[\begin{bmatrix}
+ & - & + \\
- & + & - \\
+ & - & +
\end{bmatrix}\]

Adjoint of A = transpose of the cofactor matrix \[\mathrm{adj}A=\left[A_{ij}\right]^T\]

• \[\mathrm{adj}(kA)=k^{n-1}\mathrm{adj}(A)\]

• A(adj A) = (adj A)A = ∣A∣I

• ∣adjA∣ = ∣A∣ⁿ⁻¹(for an n×n non-singular matrix)

• adj(AB) = (adjB)(adjA)

\[A=
\begin{bmatrix}
a & b \\
c & d
\end{bmatrix}\]

\[A^{-1}=\frac{1}{ad-bc}
\begin{bmatrix}
d & -b \\
-c & a
\end{bmatrix}\] if ad − bc ≠ 0

\[A^{-1}=\frac{1}{|A|}(\operatorname{adj}A)\], if ∣A∣ ≠ 0

Properties:

• \[(AB)^{-1}=B^{-1}A^{-1}\]

• \[(A^{-1})^{-1}=A\]

• \[(A^{\prime})^{-1}=(A^{-1})^{\prime}\]

• If inverse exists, it is unique.

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

Comparable Matrices

• Two matrices are said to be comparable if they have the same order
  (same number of rows and columns).

Equal Matrices

Two matrices A= [a_ij] and B=[b_ij] are equal if:

1. They are comparable (same order), and

2. Their corresponding elements are equal.

1. Write AX = B

2. Apply row operations on A
   (Same operations on B)

3. Reduce A to triangular/identity form

4. Solve equations

• Aⁿ is defined only when A is a square matrix.

• A^mAⁿ = A^m+n

• $I$

Matrix Form: AX = B

Condition:

• A must be square

• ∣A∣ ≠ 0 (Non-singular)

Formula:

\[X=A^{-1}B\]​