## Singular matrix in Discrete mathematicsWe can find that the given matrix is singular or non-singular with the help of finding the determinant of the matrix. With the help of symbol |A| or det A, we can indicate the determinant of matrix A. The matrix will be known as the singular matrix if the determinant of this matrix is equal to 0. In this section, we will learn about singular matrices, identifying a singular matrix, their properties, the difference between singular matrices and non-singular matrices, examples and many more things. ## What is Singular matrixIf the determinant of the given matrix is equal to 0, then the singular matrix must be a square matrix. In other words, a square matrix A will be known as singular if det A = 0. We have learned that we can find the inverse of a matrix A with the help of formula A From the above explanation, it is clear that a square matrix A will be known as singular if it contains following properties: - det A = 0 (we can also write it as |A| = 0)
- A
^{-1}is NOT defined (i.e., A is non-invertible)
## Identifying a Singular matrixWe have to check two conditions to identify whether the given matrix is a singular matrix, which is described as follows: - First, we will check whether the given matrix is a square matrix.
- After that, we will check whether det A = 0.
Now we will show some examples and determine whether the given matrix is a singular matrix. Suppose there is a matrix A where The order of this matrix is 2∗2. So this matrix is a square matrix. Now we will find the determinant of this matrix in the following way: |A| or det A = 3∗4 ' 6∗2 = 12 ' 12 = 0 The determinant of this matrix is 0. Hence this matrix is a square matrix. ## Steps to check the singular matrixIf we want to know whether the given matrix is singular or not, for this, we have to first calculate the determinant of the matrix. Here we will check it for the 2∗2 matrix and 3∗3 matrix one by one. ## For 2∗2 matrix:In the following steps, we will see how we can check whether the 3∗3 matrix is singular or not.
Now we will understand all these steps with the help of an example. In this example, we have a 2∗2 matrix, A where This matrix is a square matrix. So we will first calculate the determinant of the above matrix A like this: |A| or det A = 3∗4 ' 6∗2 |A| = 12 ' 12 |A| = 0 Here the determinant of A is equal to 0. Hence matrix A is a singular matrix. ## For 3∗3 matrix:In the following steps, we will see how we can check whether the 3∗3 matrix is singular or not.
Now we will understand all these steps with the help of an example. In this example, we have a 3∗3 matrix, A where This matrix is a square matrix. So we will first calculate the determinant of the above matrix A like this: |A| or det A = 0 The determinant of A is equal to 0. This is because the first and second row of this matrix is equal. Hence matrix A is a singular matrix. ## Properties of Singular matrixHere we will define some properties of a singular matrix on the basis of its definitions. These properties are described as follows: - If there is a singular matrix, then it must be a square matrix.
- In case of a singular matrix, if we calculate the determinant, then it must be zero.
- The formula to determine the determinant of a matrix with an order 2∗2 will be |A| = ad ' bc.
- The formula to determine the determinant of a matrix with an order 3∗3 will be |A| = a1(b2c3 ' b3c2) ' a2(b1c3 ' b3c1) ' a3(b1c2 ' b2c1).
- We cannot define the inverse of a singular matrix, and because of this reason, the singular matrix is non-invertible.
- If there is a null matrix with any number of order, then it must be a singular matrix.
- In case of a singular matrix, all rows and columns will not be linearly independent.
- In case of a singular matrix, the rank of this matrix must be less than its order. For example: If there is a singular matrix with order 3∗3, then its rank will be less than 3.
- In case of the matrix, there are some properties of determinants, which are described as follows:
- If there is a matrix in which any two rows or any two columns are similar, then the determinant of this type of matrix will be 0. Hence it will be a singular matrix.
- If there is a matrix in which all the elements of rows and columns are zero, then the determinant of this type of matrix will be 0. Hence it will be a singular matrix.
- The determinant of a matrix will be 0 if one of the rows (columns) is a scalar multiple of the other row (column). Hence the matrix will be a singular matrix.
## Singular matrix and Non-singular matrixAccording to the name, the non-singular matrix will not be singular. Thus, in case of the non-singular matrix, the determinant will be non-zero numbers. In other words, we can say that if det A ? 0, then the square matrix A will be known as the non-singular matrix. We can define the inverse of a matrix (A
## Theorem to Generate Singular matricesIn case of a singular matrix, there is one important theorem that is used to generate a singular matrix. According to this theorem, if there are two matrices A and B where A = [A] - The multiplication of two matrices AB will always be singular where matrix A is in order n∗1 and matrix B is in order 1∗n.
- The multiplication of two matrices AB will also be a singular where matrix A is in order n∗2 and matrix B is in order 2∗n.
With the help of this theorem, we can multiply two randomly generated matrices, which contain an order n∗k and k∗n and can generate a singular matrix. Here n > k. ## Examples of Non-singular matrixThere are a lot of examples of non-singular matrices, and some of them are described as follows:
= -7∗4 ' 2∗(-14) = -28 + 28 = 0 The determinant of this matrix is 0. Hence this matrix is a square matrix. Now we will find the determinant of matrix (b). = 2(0-6) + 1(1+12) + 3(3-0) = -12 + 13 + 9 = 10 ≠ 0 So the determinant of this matrix is not equal to 0. Hence this matrix is not a singular matrix.
det A = 1[(5∗0) ' (4∗2)] - 0[(0∗0) ' (2∗ -1)] + (-3)[(0∗4) ' (-1∗5)] |A| = (1 ∗ -8) ' 0 + (-3 ∗ 5) |A| = -8 ' 15 |A| = -23 ≠ 0 So the determinant of this matrix is not equal to 0. Hence this matrix is not a singular matrix.
(2∗k) ' (-4∗5) = 0 2k + 20 = 0 2k = -20 k = -20 /2 k = -10 Hence the value of k will be -10 for A to be singular.
(x+1) (0-1) ' x(x+3-4) + 2(1-0) = 0 -x - 1 ' x -x x x = ±1 Hence x = 1 or -1 for A to be singular.
2x + y + 2z = 3 x + z = 5 4x + y + 4z = 7
The system will have a unique solution if and only if the determinant of A is not equal to 0 (that means if A is non-singular). So we will determine the determinant of A like this: = 2(0-1) - 1(4-4) + 2(1-0) = -2 + 0 + 2 = 0 The determinant of matrix A is 0. So the system can either have no solution or have infinite solutions. Hence the system will NOT have a unique solution.
(9∗ -2) ' (6∗b) = 0 -18 ' 6b = 0 -6b = 18 b = 18 /-6 b = -3 Hence the value of b will be -3 for the given matrix A to be singular.
We can determine the inverse of matrix P with the help of following formula for the non-singular matrix: P So to find it, we will determine the determinant of this matrix P like this: |P| = (-3 ∗ -8) ' (6∗4) |P| = 24 ' 24 |P| = 0 Since we can see that the determinant of matrix P is equal to 0. So it is a singular matrix, and the inverse of this matrix does not exist. |