Before we start with the concept of a Singular and non-singular matrix, let’s first understand what is a matrix. A rectangular arrangement of objects (numbers or symbols or any other objects) is called a matrix (plural: matrices). A matrix is only a representation of the symbol, number, or object. It does not have any value. Usually, a matrix is denoted by capital letters. Matrix of order m × n, (read as m by n matrix) means that the matrix has m number of rows and n number of columns. In real life, we use Singular and non-singular matrices to solve systems of linear equations, quantum mechanics, and optics.
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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 twenty-six questions have been asked on this topic in Jee mains (2013 to 2023), two in 2013, two in 2014, two in 2015, one in 2016, six in 2019, two in 2020, four in 2021, three in 2022, and three in 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|.
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 2nd with the -ve sign and 3rd again +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 an element along a column delete their row and column compute the determinant of left 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.
A square matrix is called a singular matrix if its determinant is 0 otherwise it is called a non-singular matrix. Let's say A is a square matrix then it is singular if |A| = 0, otherwise, it will be non-singular if |A| ≠ 0.
If A and B are square matrices of same order:
i) The determinant of the transpose of matrix A is equal to the determinant of matrix A. det (A’) = det A
ii) The product of the determinant of matrices AB is equal to the product of the determinant of individual matrices. det (AB) = det (A) det (B)
iii) The determinant of the skew-symmetric matrix of odd order is zero. if A is the skew-symmetric matrix of odd order then |A| = 0.
iv) The determinant of the skew-symmetric matrix of even order is a perfect square. if A is a skew-symmetric matrix of even order then |A| is a perfect square.
v) |kA| = kn |A|, where n is the order of matrix A and k is a constant.
vii) |An| = |A|n, where n belongs to N.
viii) |AB| = |BA|
In summary, singular matrices are fundamental in linear algebra because they cannot be inverted due to having a determinant of zero. This property means they often represent systems of equations with either no solution or infinitely many solutions. They are crucial in various fields such as statistics, physics, and engineering, where understanding system constraints and solutions is essential.
Example 1: Let be the three A.P. with the same common difference and having their first terms as , respectively. Let be the terms of , respectively such that If , then the sum of the first 20 terms of an AP whose first term is and the common difference is , is equal to [JEE MAINS 2023]
Solution
Hence, the answer is 495.
Example 2:
Let α be a root of the equation
where a, b, and c are distinct real numbers such that the matrix
is singular. Then, the value of is [JEE MAINS 2023]
Solution:
Hence, the answer is 3
Example 3: The probability that a randomly chosen matrix with all the entries from the set of first 10 primes, is singular, is equal to : [JEE MAINS 2022]
Solution: Let the matrix be
If is singular then
Case I :
can take values.
can take the value which equals
Now for to hold
only one option for
Total such matrices
Case II :
In this case, can take values and can take values.
Now for
Hence, the answer is
Example 4:
The value of is : [JEE MAINS 2021]
Solution: Put a = 0, we get
OR
Given matrix is
Hence, the answer is -2
Example 5: If the minimum and the maximum values of the function
defined by
are and respectively, then the ordered pair is equal to: [JEE MAINS 2020]
Solution
Hence, the answer is ( -4, 0)
A square matrix is called a singular matrix if its determinant is 0. Let's say A is a square matrix then it is singular if |A| = 0
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|.
The product of the determinant of matrices AB is equal to the product of the determinant of individual matrices. det (AB) = det (A) det (B).
The determinant of the skew-symmetric matrix of odd order is zero. if A is the skew-symmetric matrix of odd order then |A| = 0. whereas, The determinant of the skew-symmetric matrix of even order is a perfect square. if A is a skew-symmetric matrix of even order then |A| is a perfect square.
The determinant of the transpose of matrix A is equal to the determinant of matrix A. det (A’) = det A
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