# Determinant Math Basic Concept

A determinant is a number associated with a matrix. Only SQUARE matrices have a determinant.

The symbol for a determinant can be the phrase “det” in front of a matrix variable, det(A); or vertical bars around a matrix, |A|

**Minors and Cofactors:**

If A is a square matrix, then the minor of entry aij is denoted by Mij and is defined to be the determinant of the sub-matrix that remains after the i number row and j number column are deleted from A.

The number (−1) i+j Mij is denoted by Cij and is called the cofactor of entry aij.

**Determinant (Using cofactor expansion):**

If *A *is an *n *× *n *matrix, then the number obtained by multiplying the entries in any row or column of *A *by the corresponding cofactors and adding the resulting products is called the ** determinant of A**, and the sums themselves are called

**. That is,**

*cofactor expansions of A*det(A) = *a*1 *j**C*1 *j *+ *a*2 *j**C*2 *j *+… + *a**nj **C**nj *[Cofactor expansion along the *j*th column] det(A) = *a**i*1*C**i*1 + *a**i *2*C**i *2 + … + *a**in**C**in *[Cofactor expansion along the *i*th row]

**Determinant of a triangular (upper or lower) matrix or diagonal matrix:**

If A is an n × n triangular (upper triangular, lower triangular) or diagonal matrix, then det(A) is the product of the entries on the main diagonal of the matrix; that is, det(A) = *a*11*a*22 *a*33…*a**nn *.

**Properties of Determinants of Matrices:**

- Determinant evaluated across any row or column is
**same**. - If all the elements of a row or column are zeros, then the value of the determinant is
**zero**. - Determinant of an Identity matrix (
*I*) is*n***1**. - If rows and columns are interchanged then value of determinant remains same (value does not change). Therefore,
**det(A) = det(A**, here^{T})**A**is transpose of matrix^{T} - If any two row or two column of a determinant are interchanged the value of the determinant is multiplied by
**-1**. - If all elements of a row or column of a determinant are multiplied by some scalar number
**k**, the value of the new determinant is**k**times of the given determinant. Therefore, if**A**be an n-rowed square matrix and K be any Then**|KA| = K**.^{n}|A| - If two rows or columns of a determinant are
**identical (same)**the value of the determinant is**zero**. - Let A and B be two matrix, then
**det(AB) = det(A)*det(B)**. - If A be a matrix then,
**|A**.^{n}| = |A|^{n} - Determinant of Inverse of matrix can be defined as
**|A**.^{-1}| = |A|^{-1} - Determinant of diagonal matrix, triangular matrix (upper triangular or lower triangular matrix) is product of element of the principal.
- In a determinant each element in any row or column consists of the sum of two terms, then the determinant can be expressed as sum of two determinants of same For example, If B is obtained by adding c-times a row of A to a different row, the
**det(B) = det(A)**. - If value of determinant
**Δ**becomes zero by substituting**x = α**, then**x-α**is a factor of**α**. - Here, cij denotes the cofactor of elements of aij in D.
- In a determinant the sum of the product of the elements of any row (or column) with the cofactors of the corresponding elements of any other row or column is.. For example, det(A) = ai1Cj1 + ai2Cj2 + ai3Cj3 +…… + ainCjn, here Cj1, Cj2, Cj3 …Cjn are cofactors along elements of jth row.

- Let
*λ*1 ,*λ*2 ,*λ*3 ,…,*λ**n*are the Eigenvalues of A (square matrix of order n). Then**det(A)****=***λ*1*λ*2*λ*3 …*λ**n*, product of Eigenvalues.

**DIFFERENCE BETWEEN MATRIX AND DETERMINANT:**

Matrix |
Determinant |
||

A matrix is a rectangular arrangement of numbers, expressions, or symbols in rows and columns enclosed by ( ), [ ], or || ||. | A determinant is a number associated with a matrix. It can be the phrase “det” in front of a matrix variable, det(A); or vertical bars around a matrix, |A| | ||

A matrix cannot be reduced to a single number. | A determinant can be reduced to a single number | ||

In a matrix, the number of rows may not be equal to the number of columns. | In a determinant, the number of rows must be equal to the number of columns. | ||

An interchange of rows or columns gives a different matrix. | An interchange of rows or columns gives the same determinant with +ve or –ve sign. | ||

Example: ( 1 2 ) |
Example: |1| |

#### Thank You!

## Leave a Reply