Revision: Matrices Maths HSC Commerce (English Medium) 12th Standard Board Exam Maharashtra State Board

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

If A = [a_ij] is an m × n matrix, then the transpose of A, denoted by A′ or A^T, is obtained by interchanging rows and columns.

\[A^T=[a_{ji}]\]

Symmetric Matrix:

A square matrix A = [a_ij] is called symmetric if:

A′ = A or a_ij = a_ij

Skew-Symmetric Matrix:

A square matrix A=[a_ij] is called skew-symmetric if:

A′ = −A or a_ij = −a_ij​

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

• Denoted by: A ∼ B

A square matrix A is invertible if there exists a matrix B such that

AB = BA = I

• B is called the inverse of A

• Inverse is denoted by A⁻¹

A rectangular arrangement of mn elements in the form of an ordered set of m rows, each row consisting of an ordered set of n elements, is called an m × n matrix (m × n is read as m by n).

• Each entry is called an element.
• Order of a matrix = number of rows × number of columns

General form:A=[a_ij​]

where

• i → row number

• j → column number

• a_ij​ → element in iᵗʰ row and jᵗʰ column

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)

Determinant of Order 2:

\[\det
\begin{bmatrix}
a & b \\
c & d
\end{bmatrix}=ad-bc\]

Determinant of Order 3:

• Computed by expansion along a row or column

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

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

Minor

Delete ith row and jth column: M_ij​

Cofactor of a_ij

A_ij = (−1)^i+j × (minor of a_ij)

Sign pattern:

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

1. Write AX = B

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

3. Reduce A to triangular/identity form

4. Solve equations

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.

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

• A^mAⁿ = A^m+n

• $I$

Theorem 1:

For any square matrix A:

• A + A′ is symmetric

• A − A′ is skew-symmetric

Theorem 2:

Every square matrix can be uniquely expressed as the sum of a symmetric and a skew-symmetric matrix.

\[A=\frac{1}{2}(A+A^{\prime})+\frac{1}{2}(A-A^{\prime})\]

Matrix Form: AX = B

Condition:

• A must be square

• ∣A∣ ≠ 0 (Non-singular)

Formula:

\[X=A^{-1}B\]​