Determinants

Determinants are the scalar quantities obtained by the sum of products of the elements of a square matrix and their cofactors according to a prescribed rule. They help to find the adjoint, inverse of a matrix. Further to solve the linear equations through the matrix inversion method we need to apply this concept. The cross-product of two vectors is remembered easily through the calculation of determinants.

In this article, let's learn more about the process of finding determinants of different orders and their properties, and we will work on a few solved examples.

Determinants are considered as a scaling factor of matrices. They can be considered as functions of stretching out and the shrinking in of the matrices. Determinants take a square matrix as the input and return a single number as its output.

Determinants Definition

For every square matrix, C = [\(c_{ij}\)] of order n×n, a determinant can be defined as a scalar value that is real or a complex number, where \(c_{ij}\) is the (i, j)^th element of matrix C. The determinant can be denoted as det(C) or |C|, here the determinant is written by taking the grid of numbers and arranging them inside the absolute-value bars instead of using square brackets.

Consider a matrix C = \(\left[\begin{array}{ll}1 & 2 \\ \\ 3 & 4\end{array}\right]\)

Then, its determinant can be shown as:

|C| = \(\left|\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right|\)

For the simplest square matrix of order 1×1 matrix, which only has only one number, the determinant becomes the number itself. Let's learn how to calculate the determinants for the second order, third order, and fourth-order matrices.

Calculating Determinant of 2x2 Matrix

For any 2x2 square matrix or a square matrix of order 2×2, we can use the determinant formula to calculate its determinant:

C = \(\left[\begin{array}{ll}a & b \\\\c & d\end{array}\right]\)

Its 2x2 determinant can be calculated as:

|C| = \(\left|\begin{array}{ll}a & b \\c & d\end{array}\right|\) = (a×d) - (b×c)

For example: C = \(\left[\begin{array}{ll}8 & 6 \\ \\3 & 4\end{array}\right]\)

Its determinant can be calculated as:

|C| = \(\left|\begin{array}{ll}8 & 6 \\3 & 4\end{array}\right|\)

|C| = (8×4) - (6×3) = 32 - 18 = 14

Calculating Determinant of 3x3 Matrix

For any 3x3 square matrix or a square matrix of order 3×3, \(C = \left[\begin{array}{ccc}a_{1} & b_{1} & c_{1} \\a_{2} & b_{2} & c_{2} \\a_{3} & b_{3} & c_{3}\end{array}\right] \), the determinant is represented as:

|C| (or) det C = \(\left|\begin{array}{ccc}a_{1} & b_{1} & c_{1} \\a_{2} & b_{2} & c_{2} \\a_{3} & b_{3} & c_{3}\end{array}\right| \)

Here are the steps in calculating the determinant of a 3x3 matrix.

• a₁ is fixed as the anchor number and the 2x2 determinant of its sub-matrix (minor of a₁).
• Similarly, calculate the minors of b₁ and c₁.
• Keep multiplying the small determinant by the anchor number and by its sign \(\left|\begin{array}{ccc}+ &-& + \\- & + & - \\+ &-& + \end{array}\right| \)
• Finally sum them up.

|C| = \(a_{1} \cdot\left|\begin{array}{ll}b_{2} & c_{2} \\b_{3} & c_{3}\end{array}\right|-b_{1} \cdot\left|\begin{array}{cc}a_{2} & c_{2} \\a_{3} & c_{3}\end{array}\right|+c_{1} \cdot\left|\begin{array}{ll}a_{2} & b_{2} \\a_{3} & b_{3}\end{array}\right|\)

|C| = \(a_{1}\left(b_{2} c_{3}-b_{3} c_{2}\right)-b_{1}\left(a_{2} c_{3}-a_{3} c_{2}\right)+c_{1}\left(a_{2} b_{3}-a_{3} b_{2}\right)\)

Let's consider this example:

\(B = \left[\begin{array}{ccc}3 & 1 & 1 \\4 & -2 & 5 \\2 & 8 & 7\end{array}\right] \)

Its determinant is calculated as:

|B| = \(\left|\begin{array}{ccc}3 & 1 & 1 \\4 & -2 & 5 \\2 & 8 & 7\end{array}\right| \)

= \(3 \cdot\left|\begin{array}{ll}-2 & 5 \\8 & 7\end{array}\right|-1 \cdot\left|\begin{array}{cc}4 & 5 \\2 & 7\end{array}\right|+1 \cdot\left|\begin{array}{ll}4 & -2 \\2 & 8\end{array}\right|\)

= 3 × ((-2)(7) - (5)(8)) -1 × ((4)(7) - (5)(2)) + 1 × ((4)(8) - (-2)(2))

= 3 × ((-14) - (40)) -1 × ((28) - (10)) + 1 × ((32) - (-4))

= 3 × (-54) -1 × (18) + 1 × (36)

= - 162 - 18 + 36

= -144

Note that we have calculated the determinant of a 3x3 matrix using the first row here. But any row/any column can be used to calculate the determinants.

Calculating the Determinant of a 4×4 Matrix

Consider the below mentioned 4x4 square matrix or a square matrix of order 4×4, the following changes are to be kept in mind while finding the determinant of a 4×4 matrix:

B = \(\left[\begin{array}{cccc}a_{1} & b_{1} & c_{1} & d_{1} \\a_{2} & b_{2} & c_{2} & d_{2} \\a_{3} & b_{3} & c_{3} & d_{3} \\a_{4} & b_{4} & c_{4} & d_{4}\end{array}\right]\)

• plus a₁ times the determinant of the 3x3 matrix obtained by deleting the row and the column containing a₁
• minus b₁ times the determinant of the 3x3 matrix obtained by deleting the row and the column containing b₁
• plus c₁ times the determinant of the 3x3 matrix obtained by deleting the row and the column containing c₁
• minus d₁ times the determinant of the 3x3 matrix obtained by deleting the row and the column containing d₁

\(\begin{align}|B| = &a_{1} \cdot\left|\begin{array}{lll}b_{2} & c_{2} & d_{2} \\b_{3} & c_{3} & d_{3} \\b_{4} & c_{4} & d_{4}\end{array}\right|-b_{1} \cdot\left|\begin{array}{ccc}a_{2} & c_{2} & d_{2} \\a_{3} & c_{3} & d_{3} \\a_{4} & c_{4} & d_{4}\end{array}\right|\\&+c_{1}\cdot\left|\begin{array}{ccc}a_{2} & b_{2} & d_{2} \\a_{3} & b_{3} & d_{3} \\a_{4} & b_{4} & d_{4}\end{array}\right|-d_{1} \cdot\left|\begin{array}{ccc}a_{2} & b_{2} & c_{2} \\a_{3} & b_{3} & c_{3} \\a_{4} & b_{4} & c_{4}\end{array}\right|\end{align}\)

We can use the method mentioned in the previous section to find the determinant of the 3×3 matrices. Here is an easy way of finding it.

We use a method called as multiplication of arrays to multiply two determinants of square matrices. Let us see the row by column multiplication rule to multiply two determinants of the square matrices A and B:

Multiplication of 2×2 Determinants

Consider two square matrices A and B of order 2×2, we first denote their respective determinants as |A| and |B| as shown below:

|A| = \(\left|\begin{array}{ll}\mathrm{a}_{1} & \mathrm{~b}_{1} \\\mathrm{a}_{2} & \mathrm{~b}_{2}\end{array}\right|\)

|B| = \(\left|\begin{array}{ll}\mathrm{p}_{1} & \mathrm{~q}_{1} \\\mathrm{p}_{2} & \mathrm{~q}_{2}\end{array}\right|\)

|A| × |B| = \(\left|\begin{array}{ll}\mathrm{a}_{1} & \mathrm{~b}_{1} \\\mathrm{a}_{2} & \mathrm{~b}_{2}\end{array}\right| \times\left|\begin{array}{cc}p_{1} & \mathrm{~q}_{1} \\p_{2} & \mathrm{~q}_{2}\end{array}\right|=\left|\begin{array}{ll}\mathrm{a}_{1} p_{1}+\mathrm{b}_{1} p_{2} & \mathrm{a}_{1} \mathrm{~q}_{1}+\mathrm{b}_{1} \mathrm{~q}_{2} \\\mathrm{a}_{2} p_{1}+\mathrm{b}_{2} p_{2} & \mathrm{a}_{2} \mathrm{~q}_{1}+\mathrm{b}_{2} \mathrm{~q}_{2}\end{array}\right|\)

Multiplication of 3×3 Determinants

Consider two matrices C and D of order 3×3, we first denote their respective determinants as |C| and |D| as shown below:

|C| = \(\left|\begin{array}{lll}a_{1} & b_{1} & c_{1} \\a_{2} & b_{2} & c_{2} \\a_{3} & b_{3} & c_{3}\end{array}\right|\)

|D| = \(\left|\begin{array}{lll}p_{1} & q_{1} & r_{1} \\p_{2} & q_{2} & r_{2} \\p_{3} & q_{3} & r_{3}\end{array}\right|\)

|C| × |D| = \(\left|\begin{array}{lll}
a_{1} p_{1}+b_{1} p_{2}+c_{1} p_{3} & a_{1} q_{1}+b_{1} q_{2}+c_{1} q_{3} & a_{1} r_{1}+b_{1} r_{2}+c_{1} r_{3} \\a_{2} p_{1}+b_{2} p_{2}+c_{2} p_{3} & a_{2} q_{1}+b_{2} q_{2}+c_{2} q_{3} & a_{2} r_{1}+b_{2} r_{2}+c_{2} r_{3} \\a_{3} p_{1}+b_{3} p_{2}+c_{3} p_{3} & a_{3} q_{1}+b_{3} q_{2}+c_{3} q_{3} & a_{3} r_{1}+b_{3} r_{2}+c_{3} r_{3}\end{array}\right|\)

These are some of the points to be remembered while multiplying two determinants:

• In order to multiply two determinants, we need to make sure that both are of the same order
• The value of the determinant does not change when rows and columns are interchanged, so we can also follow column by row, row by row, or column by column multiplication rules to multiply two determinants.

For square matrices of different types, when its determinant is calculated, they are calculated based on certain important properties of the determinants. Here is the list of some of the important properties of the determinants:

Property1: "The determinant of an identity matrix is always 1"

Consider the determinant of an identity matrix I = \(\left[\begin{array}{ll}1 & 0 \\\\0 & 1\end{array}\right]\), |I| = (1)(1) - (0)(0) = 1.

Thus, the determinant of any identity matrix is always 1.

Property 2: "If any square matrix B with order n×n has a zero row or a zero column, then det(B) = 0"

Consider the determinant of an identity matrix B,

|B| = \(\left|\begin{array}{ll} 2 & 2 \\0 & 0\end{array}\right|\)

|B| = (2)(0) - (2)(0) = 0

Here, the square matrix B has one zero row, and thus, the determinant of this square matrix becomes zero.

Property 3: "If C is upper or a lower-triangular matrix, then det(C) is the product of all its diagonal entries"

Consider an upper triangular matrix C with the diagonal entries 3, 2 and 4. The determinant |C| can be found as:

|C| = \(\left|\begin{array}{ccc}3 & 1 & 1 \\0 & 2 & 5 \\0 & 0 & 4\end{array}\right| \)

|C| = 3 × 2 × 4 = 24

Property 4: "If D is a square matrix, then if its row is multiplied by a constant k, then the constant can be taken out of the determinant"

Thus, the determinant remains the same in both cases.

Other important properties of determinants are:

• A square matrix C is considered to be invertible if and only if det(C) ≠ 0.
• If B and C are two square matrices with order n × n, then det(BC) = det(B) × det(C) = det(C) × det(B)
• The relationship between a determinant of a matrix D and its adjoint adj(D) can be shown as D × adj(D) = adj(D) × D = |D| × I. Here, D is a square matrix and I is an identity matrix.

The following rules are helpful to perform the row and column operations on determinants.

• The value of the determinant remains unchanged if the rows and columns are interchanged.
• The sign of the determinant changes, if any two rows or (two columns) are interchanged.
• If any two rows or columns of a matrix are equal, then the value of the determinant is zero.
• If every element of a particular row or column is multiplied by a constant, then the value of the determinant also gets multiplied by the constant.
• If the elements of a row or a column are expressed as a sum of elements, then the determinant can be expressed as a sum of determinants.
• If the elements of a row or column are added or subtracted with the corresponding multiples of elements of another row or column, then the value of the determinant remains unchanged.

Important Notes on Determinant:

Here is a list of a few points that should be remembered while studying determinant:

• A determinant can be considered as function that takes a square matrix as the input and returns a single number as its output.
• A square matrix can be defined as a matrix that has an equal number of rows and columns.
• For the simplest square matrix of order 1×1 matrix, which only has only one number, the determinant becomes the number itself.

☛ Related Topics:

• Matrix Calculator
• Matrix formula
• Diagonal Matrix Calculator
• Transpose Matrix Calculator

What is Determinant?

The determinant of a square matrix, C = [\(c_{ij}\)] of order n×n, can be defined as a scalar value that is real or a complex number, where \(c_{ij}\) is the (i,j)^th element of matrix C. It is denoted as det(C) or |C|, here the determinant is written by taking the grid of numbers and arranging them inside the absolute-value bars instead of using square brackets. Determinant of a square matrix \(C = \left[\begin{array}{ll} 4 & 2\\ \\ 5 & 3\end{array}\right]\) can be written as: \(|C| = \left|\begin{array}{ll} 4 & 2\\5 & 3\end{array}\right|\). It is obtained by multiplying the elements of any row or column by their corresponding cofactors and adding the products.

What Are Determinants Used For?

Determinants play an important role in linear equations where they are used to capture variables change in integers and how linear transformations change volume or area. Determinants are especially useful in applications where inverses and adjoints of matrices are used. The cross-product of two vectors is also calculated using determinants.

What Is the Determinant Formula for a 2×2 matrix?

For any 2x2 square matrix or a square matrix of order 2×2, we can use this determinant formula to calculate its determinant:

\(C = \left|\begin{array}{ll}a & b\\c & d\end{array}\right|\). The formula to calculate the 2x2 determinant is |C| = (a×d) - (b×c)

What are Examples of Determinants?

Consider the example of a square matrix D, D = \(\left[\begin{array}{ll}8 & 6 \\3 & 4\end{array}\right]\). Its determinant can be calculated as:|D| = \(\left|\begin{array}{ll}8 & 6 \\3 & 4\end{array}\right|\) |D| = (8×4) - (6×3) = 32 - 18 = 14.

Are Determinants Commutative?

Yes, multiplication of determinants is commutative and this can be well understood with this property: If B and C are two square matrices with order n × n, then det(BC) = det(B) × det(C) = det(C) × det(B).

What Are the Properties of Determinants?

Here is the list of some of the important properties of the determinants:

• The determinant of an identity matrix is always 1
• If any square matrix B with order n×n has a zero row or a zero column, then det(B) = 0.
• If C is upper-triangular or a lower-triangular matrix, then det(C) is the product of all its diagonal entries.
• If D is a square matrix, then if its row is multiplied by a constant k, then the constant can be taken out of the determinant.
• A square matrix C is considered to be invertible if and only if det(C) ≠ 0.
• If B and C are two square matrices with order n × n, then det(BC) = det(B) × det(C) = det(C) × det(B)
• The relationship between a determinant of a matrix D and its adjoint adj(D) can be shown as D × adj(D) = adj(D) × D = |D| × I. Here, D is a square matrix and I is an identity matrix.

How Do You Evaluate Determinants of a 3x3 Matrix?

Any 3×3 determinant can be evaluated in the following way:

\(C = \left[\begin{array}{ccc}a_{1} & b_{1} & c_{1} \\a_{2} & b_{2} & c_{2} \\a_{3} & b_{3} & c_{3}\end{array}\right] \)
Its determinant can be calculated as:

• a₁ is fixed as the anchor number and the 2x2 determinant of its sub-matrix which is a square matrix is calculated.
• The next anchor number is taken in order, now, it is b₁ and the small determinant is calculated, and finally, c₁ is taken as the anchor number and its 2x2 determinant is calculated.
• Alternately keep multiplying the small determinant by the anchor number and by its sign \(\left|\begin{array}{ccc}+ &-& + \\- & + & - \\+ &-& + \end{array}\right| \).
• |C| = \(\left|\begin{array}{ccc}a_{1} & b_{1} & c_{1} \\a_{2} & b_{2} & c_{2} \\a_{3} & b_{3} & c_{3}\end{array}\right| \)
  |C| = \(a_{1} \cdot\left|\begin{array}{ll}b_{2} & c_{2} \\b_{3} & c_{3}\end{array}\right|-b_{1} \cdot\left|\begin{array}{cc}a_{2} & c_{2} \\a_{3} & c_{3}\end{array}\right|+c_{1} \cdot\left|\begin{array}{ll}a_{2} & b_{2} \\a_{3} & b_{3}\end{array}\right|\)
• Finally, sum them up. |C| = \(a_{1}\left(b_{2} c_{3}-b_{3} c_{2}\right)-b_{1}\left(a_{2} c_{3}-a_{3} c_{2}\right)+c_{1}\left(a_{2} b_{3}-a_{3} b_{2}\right)\)

What Are the Rules to Perform Row and Column Operations on Determinants?

The following rules are helpful to perform the row and column operations on determinants.

• If the rows and columns are interchanged, then the value of the determinant remains unchanged
• When any two rows or (two columns) are interchanged, the sign of the determinant changes
• The value of the determinant of a matrix in which two rows/columns are equal is zero.
• If every element of a particular row or column of a matrix is multiplied by a constant, then its determinant also gets multiplied by the constant.
• If the elements of a row or a column are expressed as sums, then the determinant can be split into two/more determinants.
• If a row (or column) is multiplied by a number and the resultant elements are added to another row (or column), then there is no change in the determinant.

Where Can We Find a Determinant Calculator?

To find the determinant of a matrix, use the following calculator: Determinant Calculator. This will helps us to find the determinant of 3x3 matrix.

What is the Determinant of a Triangular Matrix?

The determinant of a triangular matrix can be found by calculating the product of all its diagonal entries. This is applicable to both upper-triangular and lower-triangular matrices.

Can Determinants Be Negative?

Determinants represent a scalar quantity that is a real number. Thus determinants can be negative. If the determinants are negative, it denotes the matrix has switched the orientation of its base vector. |-A| = (-1)ⁿ |A|. Take any positive determinant and switch over any two rows or columns of the matrix and find its determinant, which would result in negative determinant.