Overview of Matrices

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.

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

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

• A^mAⁿ = A^m+n

• $I$

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

• Denoted by: A ∼ B

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.

Matrix Form: AX = B

Condition:

• A must be square

• ∣A∣ ≠ 0 (Non-singular)

Formula:

\[X=A^{-1}B\]​

1. Write AX = B

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

3. Reduce A to triangular/identity form

4. Solve equations