The determinant is a scalar value that is a certain function of the entries of a square matrix. The determinant of a matrix A is commonly denoted det(A), det A, or |A|. Its value characterizes some properties of the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible. In real life, we use minors and cofactors to calculate the adjoint and inverse of a matrix.
JEE Main 2025: Sample Papers | Mock Tests | PYQs | Study Plan 100 Days
JEE Main 2025: Maths Formulas | Study Materials
JEE Main 2025: Syllabus | Preparation Guide | High Scoring Topics
In this article, we will cover the Singular and non-singular matrix. This category falls under the broader category of Matrices, which is a crucial Chapter in class 12 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination(JEE Main) and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more. A total of two questions have been asked on this topic in Jee mains (2013 to 2023).
The determinant of a matrix A is a number that is calculated from the matrix. For a determinant to exist, matrix A must be a square matrix. The determinant of a matrix is denoted by det A or |A|. So, The determinant is a scalar value that is a certain function of the entries of a square matrix
For 2 x 2 matrices
This same process we follow to evaluate the determinant of the matrix of any order. Notice that we start the first term with the +ve sign then the 2nd with the -ve sign and the 3rd again with the +ve sign, this sign sequence is followed for any order of matrix.
This whole process is row-dependent, the same process can be done using columns, which means we can select elements along the column and delete their row and column compute the determinant of the out matrix, and then multiply it with the element that we select. And we will get the same result as we get while doing the whole process along the row.
Minor is the value from the computation of a determinant of a square matrix, obtained by eliminating the row and the column that corresponds to the element under consideration.
Let A be a square matrix of order n (n ≥ 2) then the Minor of any element where i, j = 1,2,3 …. n is the determinant of the matrix leftover after deleting the ith row and jth column, is called the minor of the element and it is denoted by .
If we have the row and column passing through the element then the second-order determinant formed by the remaining elements is called the minor of and it is denoted by .
For example, the minor of of the matrix is
which is also the minor of determinant A if we write A in determinant form.
We can expand determinant w.r.t. any row or column. In each case, the value of the determinant is the same.
The cofactor of the matrix or determinant is the same as the minor of the matrix or determinant but the only difference is of sign, if i+j is even then cofactor = minor, if i + j is odd then cofactor = -minor,
Or we can write,
For example,
then, minor of the element a21 is and that of a32 is .
If we expand the determinant along the first row, then the value of in terms of minors is a11M11 - a12M12 - a13M13.
If we expand the determinant along the first row, then the value of in terms of co-factors is a11C11 + a12C12 - a13C13.
Thus, the products of the elements of any row or column with the corresponding cofactors are equal to the value of the determinant.
Also, the sum of the products of the elements of any row or column with the cofactors of the corresponding elements of any other row or column is zero.
1) If any two rows (or columns) of a determinant are identical (all corresponding elements are the same), then the value of the determinant is zero.
2) If all the elements in one row or column are zero then the value of determinant is zero.
3) For easier calculations, we expand the determinant along that row or column which contains the maximum number of zeros.
Example 1:
What is the minor of a23 in ? |
Solution
Minor of the element is the determinant excluding the 2nd row and 3rd column
So, minor =
Hence, the answer is -9
Example 2: What is the co-factor of a33 in ?
Solution
Minor of a13 =
Cofactor of a13 = = =
Hence, the answer is -7
A Determinant is a square array that gives a rule for the addition of the product of its elements. Understanding minors and cofactors helps us to calculate the value of determinants which enables us to calculate the inverse of the matrix. Knowledge of minors and cofactors enables one to analyze matrices effectively.
11 Oct'24 12:33 PM
11 Oct'24 12:30 PM
11 Oct'24 12:28 PM
11 Oct'24 12:26 PM
11 Oct'24 12:23 PM
11 Oct'24 12:21 PM
11 Oct'24 12:19 PM
11 Oct'24 12:17 PM
23 Sep'24 07:19 PM