Matrix

A matrix is a rectangular array comprising of rows and columns. If a matrix has "m" rows and "n" columns, the matrix is said to be a "m*n" ordered matrix.

The above figure highlights how we represent the elements in a m*n matrix. Generally, the matrix elements are in the form of a_ij where i is the row and j is the coulmn, that is, the element in 1^st row and 2^nd column is shown as a₁₂

TYPES OF MATRICES

1. Row matrix- A matrix having a single row
2. Column matrix-A matrix having a single column
3. Square matrix- A matrix having equal number of rows and columns

Trace of matrix

Sum of elemnts of square matrix along its principal diagonal is called Trace(TR) of a matrix

TR=a₁₁+a₂₂+.....a_nn for a n*n matrix

Diagonal Matrix

It is a type of matrix where all the terms other than the diagonals are 0 (diagonals might or might not be 0). The general notation for a diagonal matrix is diag(a₁₁,a₂₂,........a_nn)

Unit Matrix or Identity matrix

The matrix is made up of 1 and 0 in such a way that the elements are 1 only if i=j and otherwise, they are 0. An Identity matrix is shown as I.

Upper Triangular and Lower Triangular Matrix

A matrix whose elements are 0 below the principal diagonal is called Upper Triangular Matrix.

Similarly, a matrix whose elements are 0 above the principal diagonal is called Lower Triangular Matrix.

Transpose of a matrix

Transpose of a matrix A is shown as A^T or A'

Transpose of a matrix

Properties of Transpose

(i) (A')' = A

(ii) (A + B)' = A' + B', A and B being conformable matrices

(iii) (aA)' = aA', a being scalar

(iv) (AB)' = B'A', A and B being conformable for multiplication

Minors and Cofactors

Minor of an element in a matrix is the value of the determinant formed by the elements excluding its row and columns. Minor of "i"th row and "j"th column is shown as M_ij

The cofactor of an element is shown as C_ij = (-1^)i+j M_ij

Adjoint of a matrix

Adjoint of a matrix A is shown as adj A.

In order to find out Adjoint, first we need to find out the cofactor of each of the elements of the matrix A and then finding out the transpose of the new matrix.

A*adj A=|A| I_n where |A| is the determinant of A and I_n is the identity matrix of order n*n

Inverse of a matrix

Inverse of a matrix is shown as A^-1 and is shown as

A^-1=Adj A/|A|

DETERMINANTS

The magnitude of a specific square matrix is known as determinant and is calculated as:

Properties of Determinants

(i) If rows be changed into columns and columns into the rows, then the values of the determinant remains unaltered.

(ii) If any two row (or columns) of a determinant are interchanged, the resulting determinant is the negative of the original determinant.

(iii) If two rows (or two columns) in a determinant have corresponding elements that are equal, the value of determinant is equal to zero.

(iv) If each element in a row (or column) of a determinant is written as the sum of two or more terms then the determinant can be written as the sum of two or more determinants.

(v) If each element in any row (or any column) of determinant is zero, then the value of determinant is equal to zero.

|kA|=kⁿ|A| where k is a constant and A is a determinant of order n