# Determinants: Overview, Questions, Preparation

Determinant of a Matrix 2021 ( Determinants )

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Rachit Kumar SaxenaManager-Editorial

Updated on Aug 13, 2021 14:10 IST

## What is Determinant of a Matrix?

A matrix is a definite set of objects that is organised in rows as well as columns. These objects are called components of the Matrix. The classification of a matrix is written as number rows by several columns. In simple words, a matrix is an array of many numbers.

### Kinds of Matrices

There are various kinds of matrices.

Zero Matrix: A matrix whose all components are zero is known as a null matrix. The general sense of null is nothing that means zero.

Identity Matrix: It's a square matrix with all the entries zero, except for the entries on the central diagonal (upper left to lower right), which are 1.

Symmetric Matrix: Matrix is symmetric if the m-th row and n-th column element are equal to the n-th row and the mm-th column components. You could say the rows of a symmetric matrix are always the same as the columns.

Diagonal Matrix: A diagonal matrix is a kind of Matrix in which diagonal elements are non-zero and non-diagonal elements are zero.

Upper Triangular Matrix: It is considered a square matrix in which all the elements below the main diagonals are 0.

Lower Triangular Matrix: It is considered a square matrix in which all the elements above the main diagonal are 0. The definition is silent about the factor below the main diagonal.

## Determinants

Determinants are mathematical objects that are very beneficial in the study and solution of systems of linear equations. A determinant of Matrix denotes a singular number. We get this number by multiplying plus summing its elements in a particular method. Generally, a determinant is a square arrangement of numbers (written in a vertical line array) representing a definite sum of products.

Example: 4×7− 9×5 = 28 − 35 = −17

### Properties of determinants

There are several attributes of determinants, particularly the conversion of the row and column conversions, that can interpret the evaluation of any determinant significantly:

Property 1: The determinant's value is unchanged when rows switch into columns and columns into rows.

Property 2: When two rows or columns of a determinant reciprocates, the sign of the determinant changes. However, its magnitude continues alike.

Property 3: A determinant having two rows or two columns identical has the value of zero

Property 4: Multiplying all the elements of a row (or column) by a scalar (a real number) is equal to multiplying the scalar's determinant.

Property 5: A determinant sorts into a sum of two determinants along any row or column.

## Weightage of Determinant of a Matrix in Class 12

This concept is taught in chapter Determinants. The weightage of this chapter is 6 marks.

## Illustrated Examples on Determinants of Matrix

1.Evaluate

Solution.
= 2(2) – 4(–1) = 4 + 4 = 8.

2. Evaluate

Solution.

x (x) – (x + 1) (x – 1) = x2 – (x2 – 1) = x2 – x2 + 1 = 1

3. Evaluate

= 4 (–1 – 12) – 0 + 0 = – 52

## FAQs on Determinants of a Matrix

Q: What is a matrix?

A: A matrix is a factor formed by columns and rows, and the numbers are the coefficients of an equation system.

Q: What is a square matrix?

A: A square matrix is a sort of matrix of the exact number of rows and columns.

Q: What are determinants?

A: A determinant is just a specific number you comprehend when you do any math on a square matrix's values.

Q: What are the objectives of a determinant of a matrix?

A: The determinant is beneficial for determining linear equations, arresting linear change area or volume, including changing variables in integrals.

Q: What is the formula of determinant?

A: The determinant is |A| = ad − bc or a × d minus b × c.

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