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Singular Matrix - Meaning, Example, Order, Types, Determinant and Rank of Singular Matrix

 

MATRIX: Any set of numbers or functions which are arranged in a row and column type format in order to form a rectangular array is called a matrix. Those functions or numbers are called elements of the matrix.

Matrices are used widely in different branches of mathematics. Using matrices makes our workflow easy.

Let’s look at a small example to understand a matrix. Twinkle had 5 pencils and 6 erasers, Pinku had 6 pencils and 7 erasers, and Kenny had 7 pencils and 8 erasers.

Now, we can arrange the above elements in a matrix form, i.e., in a row and column format. Let us put pencils in a row and erasers in a column; then this can be expressed as,

image

Therefore, the above arrangement is known as a matrix.

ORDER OF A MATRIX

Every matrix has an order. It's nothing but an indication of how many rows and columns are arranged in a matrix. Let m be the number of rows and n be the number of columns, then m × n is the order of the matrix. From the above example,

The order of the matrix = 3 × 3.

TYPES OF MATRICES

  1. Singular matrix
  2. Square matrix
  3. Row matrix
  4. Diagonal matrix
  5. Column matrix, etc.

DETERMINANT

A determinant is the scalar value of all the entries in a square matrix. Only square matrices have determinants. It is written as det(A) or ⏐A⏐. It is important to understand determinants because operations on deteminants are somewhat similar as compared to matrices.

For a 2 ×2 matrix, the determinant can be defined as,

image2

det(A) = a (cb – a2) – b (b2 – ac) + c (ba – c2)

Until now, we have covered all the topics related to matrices, and hence it is easy to understand our main topic, singular matrix.

SINGULAR MATRIX

If the determinant of a square matrix is 0, then the matrix is said to be a singular matrix.

We know the inverse of a matrix (A') = adjoint(A)/A

Since ⏐A⏐of a singular matrix is 0, therefore the inverse of the matrix doesn't exist for a singular matrix.

The multiplicative inverse also doesn't exist in singular matrices.

If ⏐A⏐= 0, it is a singular matrix.

If ⏐A⏐≠ 0, it is a non – singular matrix.

RANK OF SINGULAR MATRIX

The rank of a singular matrix must be less than the minimum, i.e., less than the number of rows and number of columns. Since the rows of a singular matrix are not linearly independent, therefore say if the order of the matrix A is 3 × 3,

Then the rank of A ≤ 3.

EXAMPLE: Find whether the given matrix A is singular

or not?

image3

SOLUTION:⏐A⏐ = (4 × 2) – (1 × 8)

= 8 – 8⇒ 0

We know that if ⏐A⏐= 0, then the matrix is a singular matrix. Therefore, the given matrix is singular.

EXAMPLE: Find the inverse of

image4

SOLUTION :⏐A⏐= 2(0 – 1) – 1 (4 – 4) +2 (1 – 0)

image5

Hence, ⏐A⏐= 0, the inverse does not exist. Hence, the given matrix is a singular matrix.

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