Determinant of Matrix

The determinant of a matrix is a function that maps every square matrix to a unique number (real number or complex number). If A is the set of all square matrices (of all orders) and B is the set of all numbers (both real and complex) then the determinant function f is f : A → B and is defined as f(x) = y, where 'y' is the determinant of matrix 'x'.

Let us learn the process of finding determinant of the matrix for matrices of orders 1x1, 2x2, 3x3, etc along with a few examples. Also, let us focus on the properties of determinants.

The determinant of matrix is the sum of products of the elements of any row or column and their corresponding co-factors. The determinant of matrix is defined only for square matrices. For any square matrix A, the determinant of A is denoted by det A (or) |A|. It is sometimes denoted by the symbol Δ. The process of calculating the determinants of 1x1 matrices and 2x2 matrices is pretty simple whereas the process becomes more complex as the order of the matrix increases. The process of finding the determinant of matrix involves minors and co-factors. Let us first recall how to find the minors and co-factors of the elements of a matrix.

Determinant of a 2x2 Matrix

The determinant of a 2x2 matrix A = \(\left[\begin{array}{cc}a & b \\ \\ c & d\end{array}\right]\) is |A| = ad - bc. It is simply obtained by cross multiplying the elements starting from top left and then subtracting the products.

Minor of Element of a Matrix

The minor of an element \((a_{ij})\) of a square matrix of any order is the determinant of the matrix that is obtained by removing the row (i^th row) and the column (j^th column) containing the element. We can understand this by an example.

Example: For a matrix A = \(\left[\begin{array}{ccc}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{array}\right]\),

• The minor of 6 is,
  \(\left|\begin{array}{ccc}1 & 2 & ̶3̶ \\ ̶4̶ & ̶5̶ & { ̶6̶} \\ 7 & 8 & ̶9̶\end{array}\right|\) = \(\left|\begin{array}{cc}1 & 2 \\ \\ 7 & 8\end{array}\right|\)
  = 1(8) - 2(7) = 8 - 14 = -6.
• The minor of 9 is,
  \(\left|\begin{array}{ccc}1 & 2 & ̶3̶ \\ 4 & 5 & ̶6̶ \\ ̶7̶ & ̶8̶ & ̶9̶\end{array}\right|\) = \(\left|\begin{array}{cc}1 & 2 \\ \\ 4 & 5\end{array}\right|\)
  = 1(5) - 2(4) = 5 - 8 = -3.

Co-factor of Element of a Matrix

The cofactor of an element \(a_{ij}\) of a square matrix of any order is its minor multiplied by (-1)^{i + j}. i.e.,

• Co-factor of an element = (-1)^{row number + column number} (minor of the element)

We found the minors of elements 6 and 8 in the previous example. Let us calculate the cofactors of the same elements now.

Example: For the same matrix A = \(\left[\begin{array}{ccc}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{array}\right]\),

• Since 6 is in the 2^nd row and 3^rd column,
  co-factor of 6 = (-1)²⁺³ (minor of 6) = (-1)⁵ (-6) = 6
• Since 9 is in the 3^nd row and 3^rd column,
  co-factor of 9 = (-1)³⁺³ (minor of 9) = (-1)⁶ (-3) = -3

The co-factors of elements of any matrix are nothing but the minors but multiplied by the alternative + and - signs (beginning with + sign for the first element of the first row).

In the previous section, we have seen that the determinant of matrix is the sum of products of elements of any row (or any column) and their corresponding cofactors. Thus, here are the steps to find the determinant of matrix (a 3×3 matrix or any other matrix).

• Step 1: Choose any row or column. We usually choose the first row to find the determinant.
• Step 2: Find the co-factors of each of the elements of the row/column that we have chosen in Step 1.
• Step 3: Multiply the elements of the row/column from Step 1 with the corresponding co-factors obtained from Step 2
• Step 4: Add all the products from Step 3 which would give the determinant of the matrix.

Example: Use the above steps to compute the determinant of 3x3 matrix A = \(\left[\begin{array}{ccc}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{array}\right]\).

Solution:

Step 1: We choose the first row with elements 1, 2, and 3.

Step 2: Let us compute the cofactors of these elements:

Co-factor of 1 = (-1)¹⁺¹ Minor of 1 = (-1)² \(\left|\begin{array}{cc}5 & 6 \\ \\ 8 & 9\end{array}\right|\) = 5(9) - 6(8) = -3
Co-factor of 2 = (-1)¹⁺² Minor of 2 = (-1)³ \(\left|\begin{array}{cc}4 & 6 \\ \\ 7 & 9\end{array}\right|\) = -1 (4(9) - 6(7)) = -1(-6) = 6
Co-factor of 3 = (-1)¹⁺³ Minor of 1 = (-1)⁴ \(\left|\begin{array}{cc}4 & 5 \\ \\ 7 & 8\end{array}\right|\) = 4(8) - 5(7) = -3

Step 3: Multiply the elements by their cofactors.

1(co-factor of 1) = 1 (-3) = -3
2(co-factor of 2) = 2(6) = 12
3(co-factor of 3) = 3(-3) = -9

Step 4: Add them to get the determinant.

det A = -3 + 12 - 9 = 0.

All these steps can be summarized in a single step as follows:

det A = 1(co-factor of 1) + 2(co-factor of 2) + 3(co-factor of 3)

= 1 \(\left|\begin{array}{cc}5 & 6 \\ \\ 8 & 9\end{array}\right|\) - 2 \(\left|\begin{array}{cc}4 & 6 \\ \\ 7 & 9\end{array}\right|\) + 3 \(\left|\begin{array}{cc}4 & 5 \\ \\ 7 & 8\end{array}\right|\)

= 1 [5(9) - 6(8)] - 2 [4(9) - 6(7)] + 3 [4(8) - 5(7)]

= 1 (-3) - 2 (-6) + 3 (-3)

= -3 + 12 - 9

= 0

Note: Here we have used a negative sign with 2 in the second step because we get a minus sign while finding the co-factor of 2.

The process of finding the determinant of matrix that is explained in the previous section can be used to find the determinant of a matrix of any order. But there are some tricks to find the determinants of 1x1, 2x2, and 3x3 matrices. These tricks are very useful as we come across finding the determinants of matrices of these orders only most of the time while solving problems.

Determinant of 1×1 Matrix

1x1 matrix is a row with just 1 row and 1 column and hence it has only one element. The determinant of any 1×1 matrix is always equal to the element of the matrix. i.e.,

• If A = [x]_1×1, then |A| (or) det A = x

Determinant of 2×2 Matrix

As we discussed earlier, its determinant is obtained by subtracting the product of elements of the non-principal diagonal from the product of the elements of the principal diagonal. i.e.,

• If A = \(\left[\begin{array}{cc}a & b \\ \\ c & d\end{array}\right]\) then |A| (or) det A = ad - bc

Determinant of 3×3 Matrix (Shortcut)

The shortcut to find the determinant of 3x3 matrix is, just write the matrix twice and apply the following trick. Here is the shortcut (easiest way) to find the determinant of 3x3 matrix A = \(\left[\begin{array}{ccc}a & b & c \\ p & q & r \\ x & y & z\end{array}\right]\).

The properties of determinants are useful in finding the determinant of a matrix without actually using the process of finding it. These are helpful in evaluating the complex determinants. These include how the determinant changes with respect to elementary row operations.

Property 1

"The determinant of a matrix is equal to the determinant of its transpose."

Example:

\(\left|\begin{array}{ccc}a & b & c \\ p & q & r \\ x & y & z\end{array}\right|\) = \(\left|\begin{array}{ccc}a & p & x \\ b & q & y \\ c & r & z\end{array}\right|\)

Try verifying it.

Property 2

"If any two rows (or columns) of a determinant are interchanged, then the sign of the determinant changes."

Example:

\(\left|\begin{array}{ccc}a & b & c \\ p & q & r \\ x & y & z\end{array}\right|\) = - \(\left|\begin{array}{ccc}p & q & r \\ a & b & c \\ x & y & z\end{array}\right|\)

Here, the first and second rows of the left side determinant are interchanged. One can easily verify this by finding both determinants.

Property 3

"If any two rows (or columns) of a determinant are identical, then the determinant is 0."

Example:

\(\left|\begin{array}{ccc}a & b & c \\ a & b & c \\ x & y & z\end{array}\right|\) = 0

Property 4

"If all elements of a row (or column) of a matrix of a determinant are zeros, then the value of the determinant is 0."

Example:

\(\left|\begin{array}{ccc}0 & 0 & 0 \\ p & q & r \\ x & y & z\end{array}\right|\) = 0

Property 5

"If each element of a row (or column) of a determinant is multiplied by a scalar k, then the value of the resultant determinant is k times the value of the original determinant."

Example:

\(\left|\begin{array}{ccc}ka & kb & kc \\ p & q & r \\ x & y & z\end{array}\right|\) = k \(\left|\begin{array}{ccc}a & b & c \\ p & q & r \\ x & y & z\end{array}\right|\)

Property 6

"If each element of a row (or column) of a determinant is expressed as sum of two (or more) numbers, then the determinant can be split into the sum of two (or more) determinants."

Example:

\(\left|\begin{array}{ccc}a + k & b + l & c + m \\ p & q & r \\ x & y & z\end{array}\right|\) = \(\left|\begin{array}{ccc}a & b & c \\ p & q & r \\ x & y & z\end{array}\right|\) + \(\left|\begin{array}{ccc}k & l & m \\ p & q & r \\ x & y & z\end{array}\right|\)

Property 7

"If each element of a row (or column) is multiplied by a constant and the elements are added to the corresponding elements of another row (or column), then the determinant remains unchanged."

Example:

\(\left|\begin{array}{ccc}a & b & c \\ p & q & r \\ x & y & z\end{array}\right|\) = \(\left|\begin{array}{ccc}a+kp & b+kq & c+kr \\ p & q & r \\ x & y & z\end{array}\right|\)

Important Notes on Determinant of Matrix:

• The determinant of an identity matrix is always 1.
• The determinant of a diagonal matrix is always the product of elements of its principal diagonal.
• The determinant of an orthogonal matrix is either +1 or -1.
• The determinant of a matrix can be either positive, negative, or zero.
• The determinant of matrix is used in Cramer's rule which is used to solve the system of equations.
• Also, it is used to find the inverse of a matrix. If the determinant of a matrix is not equal to 0, then it is an invertible matrix as we can find its inverse.
• If A is a square matrix of order 3×3, then |kA| = k³ |A|, for any scalar k.
• A square matrix A is called singular if |A| = 0 and non-singular if |A| ≠ 0.
• The determinant is helpful in computing the eigenvalues and eigenvectors of a matrix.

☛ Related Topics:

• Determinant of Matrix Calculator
• Matrix Calculator
• Matrix Multiplication Calculator
• Types of Matrices

What is the Definition of Determinant of Matrix?

The determinant of a matrix is obtained by multiplying the elements any of its rows or columns by the corresponding cofactors and adding all the products. The determinant of a square matrix A is denoted by |A| or det (A).

How to Find the Determinant of Matrix?

To find the determinant of a square matrix:

• Select any row or column
• Find the cofactors of all the elements of the row or column that you have selected
• Multiply the elements of the row or column by their corresponding cofactors
• Add the products from the last step. This sum gives the determinant.

What is the Determinant of Matrix Product?

The determinant product of two square matrices is the product of the determinants of the individual matrices. i.e., for any two square matrices A and B each of order nxn, det(AB) = det(A) × det(B).

What is the Determinant of Matrix Square?

For any two square matrices A and B of the same order, det(AB) = det A det B. So det(A²) = det(AA) = detA det A = (det A)². So the determinant of the square of a matrix is the square of the determinant of the matrix.

What are the Applications of the Determinant of Matrix?

The determinants are used to:

• find the inverse of a matrix.
• solve the system of equations
• in Cramer's rule.

What is the Determinant of Matrix When All its Elements are Equal?

Using one of the properties of determinants, when any two rows or columns of a matrix are equal, its determinant is zero. Using this, the determinant of a matrix whose all elements are equal is equal to 0.

What is the Determinant of Matrix When All its Elements are Zeros?

Using one of the properties of determinants, when all the elements of any row or column are zeros, its determinant is zero. From this, the determinant of a matrix whose all elements are zeros is equal to 0.

What is the Easiest Way of Finding Determinant of Matrix?

Here are some easiest ways/formulas to find the determinant of matrix.

• The determinant of a 1×1 matrix: If A = [x]_1×1, then |A| = x.
• The determinant of a 2×2 matrix: If A = \(\left[\begin{array}{cc}a & b \\ \\ c & d\end{array}\right]\) then |A| = ad - bc.
• The determinant of a 3×3 matrix: If A = \(\left[\begin{array}{ccc}a & b & c \\ p & q & r \\ x & y & z\end{array}\right]\) then |A| = a (qz - ry) - b (pz - rx) + c (py - qx).