Determinants: Types, Properties & Examples

Determinants are defined as a scalar value which is obtained by functions of elements of a square matrix. It helps solve various algebraic calculations in very simple ways. 

• Determinants help us determine the uniqueness of solutions in the matrix.
• It helps solve the formulation of algebraic equations.
• The method is used to represent the properties of the matrix.
• Determinants have a nonzero value when the matrix is invertible.
• It is used to determine the adjoint and inverse of the matrix.
• The process is used to analyse and solve linear equations.
• It is used in disciplines like Science, Engineering, Economics, Social Science, and others. 
• It is represented like a matrix, except it has a modulus symbol.
• The 3D effect in video games is created by using determinants.

Read More: Applications of Determinants and Matrices

Key Terms: Determinant, Matrix, Square matrix, Algebraic Equations, Complex number, Real number, Minors, Cofactors, Identity Matrix, Transpose Matrix, Inverse Matrix, Singular Matrix, Non-Singular Matrix

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What are Determinants?

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Determinants are the numerical value of a square matrix. To every square matrix, we can associate a real or complex number. This number is called the determinant of that square matrix. 

• It is used to map functions associated with a square matrix with a unique real or complex number.
• So, we can say that a square matrix A of order ‘n’ with the same number of rows and columns will have a single real or complex number.
• This number is the determinant of the square matrix A. 
• The determinant is denoted by det A or |A|.
• To find the value of a determinants correctly, it is required to make the maximum possible zero in a row or column.
• It expand the determinant that corresponds to that row or column.

Some key points to note are as follows:

• |A| should be read as determinant A and not the absolute value of A.
• A determinant expresses numerical value, but a matrix does not give numerical value.
• It will always have an equal number of rows and columns, i.e., determinants only belong to Square Matrices. 
• Thus, the determinant of a 1 × 1 matrix is that number itself.

Overview image of Determinants

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Determinants Detailed Video Explanation:

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Types of Determinants

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There are commonly three types of determinants which are as follows:

First Order Determinant

First Order Determinant is used for the calculation of the determinant for a matrix of order 1. If [a]=A, then the determinant of A will be equal to ‘a’.

Read More: Elementary Matrix Operations

Second-Order Determinant

Second-Order Determinant is used for matrices of order 2. The determinant of a matrix of order 2 can be calculated by first multiplying the diagonally opposite elements in the matrix and then finding the difference between these two products.

Third Order Determinant

Third Order Determinant is used for matrices of order 3. The determinant of a matrix of order 3 can be calculated by first adding the product of the diagonally opposite elements of the matrix and then subtracting the sum of elements perpendicular to the line segment.

The above methods are mostly used by expanding the determinant along the row of the matrix consisting the maximum number of zeroes.

Read More: Difference between rows and columns

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Properties of Determinants

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The important Properties of Determinants are as follows:

Property 1

The value of a determinant is zero if all the elements of any row or column of the determinant are zero.

Property 2

The value of a determinant is zero if all the corresponding elements of any two rows or columns of the determinant are identical or proportional.

Property 3

If any two rows or columns of a determinant are interchanged, then the determinant obtained will have a value negative of the value of the given determinant.

Read More: applications of matrices and determinants

Property 4

If any one row or column of a determinant has elements that are each a multiple of scalar k, then the determinant will have a value that is a multiple of k.

Property 5

A determinant does not change its value if its rows and columns are interchanged.

Property 6

The elements of any row or column of a determinant, when added or subtracted with the multiples of corresponding elements of any other column, do not affect the value of the determinant.

Property 7

A determinant can be expressed as the sum of two or more determinants if some or all elements of any of its rows or columns are expressed as a sum of two or more terms.

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How to calculate Determinants?

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There are three different types of determinants which first order, second order and third order determinants. For a matrix of order 1 x 1 then the value of determinant is equal to the number itself.

The process to calculate determinants of second and third order are as follows:

Calculating determinants of 2 x 2 Matrix

The square matrix of order 2 x 2 is represented as follows:

• Next use the Determinant Formula to calculate the determinants.
• |A| = (a×d) - (b×c)

Calculating determinants of 3 x 3 Matrix 

The square matrix of order 3 x 3 is represented as follows:

• Next use the Determinant Formula to calculate the determinants.
• |A| = (a×exi) + (bxfxg) + (cxdxh) – (cxexg) – (bxdxi) – (axfxh)

Read More: Matrices Ncert Solutions

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Minors and Cofactors

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Minors and Cofactors are terms commonly related to determinants. The minor of an element a_ij of a determinant is basically the determinant that is obtained after deleting its ‘i^th’ row and ‘j^th’ column in which the element a_ij lies.

• The minor of an element a_ij is denoted by M_ij.
• For a determinant with order n(n ≥ 2) is a determinant of order n – 1.
• This can be also simplified by saying that the minor of any element in a determinant is the determinant of just the single element.
• The cofactor of an element a_ij in a determinant can be defined by A_ij= (-1)^i+jM_ij

Two points to be noted about Cofactors and Minors are-

• Minors and cofactors can be used to expand determinants accordingly.
• If the elements of a row or column are multiplied with the cofactors of any other row or column, their resultant sum is zero.

Minors and Cofactors

Read More: Operations on Matrices

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Things to Remember

• Determinants are the numerical value of a square matrix.
• To every square matrix, we can associate a real or complex number.
• It can be thought of as a mapping function that associates a square matrix with a unique real or complex number.
• There are three common types of determinants, namely first-order determinant, second-order determinant, and third-order determinant. 
• To find the value of a determinant correctly, it is required to make the maximum possible zero in a row or column.
• It is done using certain properties and then expanding the determinant corresponding to that row or column.

Read More: Matrix Multiplication

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Previous Years Questions

• If ω to is a complex cube root of unity, then ∣∣ ∣ ∣∣1ωω2ωω21ω21ω∣∣ ∣ is equal to...[BITSAT 2006]
• The value of ∣∣ ∣∣123−4362−79∣∣ is….[BITSAT 2010]
• If A=⎡⎢⎣1−1121−3111⎤⎥⎦,10B⎡⎢⎣422−50α1−23⎤⎥⎦A=[1−1121−3111],10B[422−50α1−23] and BB is the inverse of AA, then the value of αα is...[KCET 2007]
• The roots of the equation |x−1111x−1111x−1|=0 are​
• If t55, t1010 and t2525 are 5thth, 10thth, and 25thth terms of an A.P. respectively, then the value of ∣∣ ∣∣t5t10t2551025111∣∣ ∣∣|t5t10t2551025111| is equal to
• If A=[1235],A=[1235], then the value of the determinant |A2009−5A2008||A2009−5A2008| is
• If ∣∣ ∣∣3i−9i129i−1109i∣∣ ∣∣=x+iy|3i−9i129i−1109i|=x+iy, then​
• If A=[logx−1−logx2]A=[logx−1−logx2] and if det(A)=2det(A)=2, then the value of xx is equal to​
• If l,m and n are real numbers such that l2+m2 +n2=0, then |1+l2lmlnlm1+m2mnlnmn1+n2| is equal to
• If ω≠1ω≠1 is a cube root of unity, then the value of ∣∣ ∣ ∣∣1+2ω100+ω2001ωω21+ω100+2ω200ω21ω1+ω100+2ω200∣∣ ∣ ∣∣|1+2ω100+ω2001ωω21+ω100+2ω200ω21ω1+ω100+2ω200| is equal to
• Let Δ=∣∣ ∣ ∣∣1111−1−w2w21ww4∣∣ ∣ ∣∣Δ=|1111−1−w2w21ww4|, where w≠1w≠1 is a complex number such that w3=1w3=1. Then ΔΔ equals
• The value of the determinant ∣∣ ∣∣15!16!17!16!17!18!17!18!19!∣∣ ∣∣|15!16!17!16!17!18!17!18!19!| is equal to​
• If A=⎡⎢⎣1−570791189⎤⎥⎦A=[1−570791189], then trace of matrix AA is​
• Let A=(x+23x3x+2),B=(x05x+2)A=(x+23x3x+2),B=(x05x+2). Then all solutions of the equation det (AB)=0(AB)=0 is​
• If the rank of the matrix ⎡⎢⎣−1252−4a−41−2a+1⎤⎥⎦[−1252−4a−41−2a+1] is 1, then the value of aa is
• If f:[0,π/2)→Rf:[0,π/2)→R is defined as f(θ)=∣∣ ∣∣1tanθ1−tanθ1tanθ−1−tanθ1∣∣ ∣∣f(θ)=|1tan⁡θ1−tan⁡θ1tan⁡θ−1−tan⁡θ1|. Then, the range of ff is​
• If ωω is an imaginary cube root of unity, then the value of the determinant​
• If the system of equations x+ky−z=0,3x−ky−z=0x+ky−z=0,3x−ky−z=0 and x−3y+z=0x−3y+z=0, has non-zero solution, then kk is equal to​