Rank of Matrix (2nd Sem)

 

Chapter

Rank of Matrix

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Rank of Matrix : Definition:

A number $r$ is said to be the rank of a matrix $A$ if it possesses the following two properties:

(i) There is at least one square submatrix of $A$ of order $r$ whose determinant is not equal to zero.

(ii) If the matrix $A$ contains any square submatrix of order $r + 1$, then the determinant of every square submatrix of $A$ of order $r + 1$ should be zero.

In short, the rank of a matrix is the order of any highest order non-vanishing minor of the matrix.

Thus, the rank of a matrix $A$ is the order of any highest order square submatrix of $A$ whose determinant is not equal to zero.

We shall denote the rank of a matrix A by the symbol ρ(A).

It is obvious that the rank $r$ of an $(m \times n)$ matrix can at most be equal to the smaller of the numbers $m$ and $n$, but it may be less.

If there is a matrix $A$ which has at least one non-zero minor of order $n$ and there is no minor of $A$ of order $n + 1$, then the rank of $A$ is $n$. Thus, the rank of every non-singular matrix of order $n$ is $n$. The rank of a square matrix $A$ of order $n$ can be less than $n$ if and only if $A$ is singular, i.e., $|A| = 0$.

Note 1:

Since the rank of every non-zero matrix is $\geq 1$, we agree to assign the rank, zero, to every null matrix.

Note 2:

Every $(r + 1)$-rowed minor of a matrix can be expressed as a linear combination of its $r$-rowed minors.
Therefore, if all the $r$-rowed minors of a matrix are equal to zero, then obviously all its $(r + 1)$-rowed minors will also be equal to zero.

Important:

The following two simple results will help us very much in finding the rank of a matrix:

(i) The rank of a matrix is $\leq r$, if all $(r + 1)$-rowed minors of the matrix vanish.

(ii) The rank of a matrix is $\geq r$, if there is at least one $$r-rowed minor of the matrix which is not equal to zero.

Q1: Find the rank of the following matrices:

$$A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 1 & 0 \\ 0 & 1 & 2 \end{bmatrix}$$

Answer : For the solution, see the video given below (हल के लिए, नीचे दी गई विडियो देखे)

​Q2: Find the rank of the following matrices:

$$A=\left[\begin{array}{ccc}1& 2& 3\\ 2& 4& 5\end{array}\right]$$

Answer : For the solution, see the video given below (हल के लिए, नीचे दी गई विडियो देखे)

Q3: Under what condition the rank of the following matrix is 3

$$A=\left[\begin{array}{ccc}2& 4& 2\\ 2& 1& 2\\ 1& 0& x\end{array}\right]$$

Answer : For the solution, see the video given below (हल के लिए, नीचे दी गई विडियो देखे)

$$<h1\; style="text-align:\; center;">Lecture\; 1</h1>$$

Lecture 1 PDF Download from this link

Answer : For the solution, see the video given below (हल के लिए, नीचे दी गई विडियो देखे)

Q4: Find which value of b the rank of the matrix

$$A = \begin{bmatrix} 1 & 5 & 4 \\ 0 & 3 & 2 \\ b & 13 & 10 \end{bmatrix}$$is 2?

Answer : For the solution, see the video given below (हल के लिए, नीचे दी गई विडियो देखे)

Q5: Find the values of a so that rank(A)<3, where A is the matrix

$$A = \begin{bmatrix} 3a - 8 & 3 & 3 \\ 3 & 3a - 8 & 3 \\ 3 & 3 & 3a - 8 \end{bmatrix}$$

Q5: Find the values of a so that rank(A)<3, where $A$ is the matrix

$$A=\left[\begin{array}{ccc}\mathrm{3a-8}& 3& 3\\ 3& \mathrm{3a-8}& 3\\ 3& 3& \mathrm{3a-8}\end{array}\right]$$

Answer : For the solution, see the video given below (हल के लिए, नीचे दी गई विडियो देखे)

Q6: Prove that the points $(x_1, y_1), (x_2, y_2)$ and $(x_3, y_3)$ are collinear if and only if the rank of the matrix

$$A=\left[\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right]$$

Answer : For the solution, see the video given below (हल के लिए, नीचे दी गई विडियो देखे)

Q7: Determine the rank of matrix:

$$\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \end{bmatrix}$$

Answer : For the solution, see the video given below (हल के लिए, नीचे दी गई विडियो देखे)
$$$$

Q8: Determine the rank of each of matrices:

(i) $\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$

(ii) $\begin{bmatrix} 5 & 10 \\ 3 & 6 \end{bmatrix}$

$$Answer : For the solution, see the video given below (हल के लिए, नीचे दी गई विडियो देखे)

Lecture 2

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