LINEAR ALGEBRA ENGINEERING MATHEMATICS GATE 2020 STUDY MATERIAL PDF CIVILENGGFORALL

Linear Algebra
Determinants
Minor
Co-factor
Algebra of Matrices
Equality of Two Matrices
Transpose of a Matrix
Orthogonal Matrix
Rank of a Matrix
Cramer’s Rule
Eigen Values and Eigen Vectors

Determinants : A determinant of order n has n rows and n columns. It has n x n elements.

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MINOR : If a_ij is an element which is in the i^th row and j^th column of a square matrix A, then determinant of the matrix obtained by deleting i^th row and j^th column of A is called minor of a_ij and is denoted by M_ij

Properties of Determinants

If all the elements of one column or one row of a determinant are multiplied by the same number “ C”, then value of the new determinant is “C” times the value of the given determinant.
Value of a determinant does not change when rows and columns are interchanged.
If any two columns or two rows of a determinant are interchanged, then value of the determinant is multiplied by – 1.
In a determinant, sum of the products of the element of any column (or row) with the cofactors of the corresponding elements of any other column (or row) is zero.

If two columns (or two rows) of a determinant are identical, then value of the determinant is zero.
If in a determinant each element in any column (or row) consists of sum of two terms, then determinant can be expressed as the sum of two determinants of the same order.
If the elements of a column (or row) of a determinant are added K times the corresponding elements of another column (or row), then value of the determinant obtained is equal to value of the original determinant.

CO-FACTOR : If a_ij is an element which is in the i^th row and j^th column of a square matrix A, then product of (– 1)^{i + j} and minor of a_ij is called cofactor of a_ij and is denoted by A_ij

Elementary Transformation of a Matrix

Following operations on a matrix are called elementary transformations.

Interchange of any two rows (or columns).
Multiplication of elements of any row (or column) by any non-zero number.
Addition to elements of any other row (or column) the corresponding elements of any other row (or column) multiplied by any number.