Determinant of a square matrix
Every square matrix can be associated with a real number called determinant.
The determinant of a matrix A is written as |A| or det A.
Finding the determinant depends on the dimension of the matrix A.
Determinants exist only for square matrices.
Determinant of a 2 X2 matrix
A= , |A|=det A=det =ad-bc,(multiply the leading diagonal)
Note: the vertical bars, when used in the context of matrices, represents determinants and not absolute value.
Example: find the determinant of the matrix
det( A) = |A|= (-6x 1) – (2x7)=-6-14=-20
Minors and Cofactors:
If A is a square matrix, the minor Mij of the entry aij is the determinant of the matrix obtained by deleting the ith row and the jth column of A.
In other words, the minor of an element is the value of the determinant found by deleting the row and column of that element.
Example: Find the minor of 1 of the matrix C =
The minor of 1 is
= 15 - 8 = 7
Definition: If A is a square matrix, with minor Mij of the entry aij , then the cofactor Cij of the entry aij is
Cij = (-1)i+j Mij
Example: In the above example the cofactor of 1 is C11 = (-1)1+1 M11 =(-1)2 7 = 7
Sign pattern for Cofactors:
The + and – signs represent (-1)i+j for each element.
Using Cofactors to find the determinant of a square matrix:
|A| = a11C11 + a12C12 +… + a1nC1n
Example: Find the determinant by expanding along first row of matrix A =
The minor of 1 is 7 and the cofactor is 7.
The minor of 0 is = -51 and the cofactor is 51.
The minor of -3 is = -52 and the cofactor is -52.
Then det (A) = 1(7) + 0(51) +(-3) (-52) = 163.
Properties of Determinants:
|Courses/Topics we help on|
|Discrete Mathematics||Applied Calculus I||Applied Calculus II|
|Healthcare Statistics and Research||Advanced Engineering Mathematics I
||Advanced Engineering Mathematics II|
|Introduction to Algebra||Basic Algebra||Algebra for College Students|
|Algebra for College Students||Pre-Calculus||Statistics for Decision-Making|
|Polar Co-ordinates||Area in Polar Coordinates||Solving Systems of Equations|
|Systems of Inequalities||Quadratic Equations||Matrices and System of Equations|
|The Determinant of a Square Matrix||Cramer's Rule||Ellipse|
|Hyperbola||Rate of Change||Measurement of Speed|
|Finding Limits Graphically||Higher Order Derivatives||Rolle's Theorem and Mean Value Theorem|
|Concavity and Second Derivative Test||Limits at Infinity||Indefinite Integration|
|Definite Integration||Integration by Substitution||Area of a Region Between Two Curves|
|Volume by Shell Method and Disc Method||Integration by Parts||Trigonometric Integration|
|Differential Equations||Slope Fields||Growth and Decay|
|System of Differential Equations||Parametric Equations||Complex Numbers|
|The Inverse of a Square Matrix||Parabola||Functions and Their Graphs|
|Evaluating Limits Analytically||Increasing and Decreasing Functions||Newton's Method|
|Finding Area Using Integration||Numerical Integration||Moments|
|Partial Fractions||Separation of Variables||Second Order Differential Equations|