Chapter 3: 15P (page 88)

Find the rank of each of the following matrices.
$(\begin{matrix} 1 & 1 & 2 \\ 2 & 4 & 6 \\ 3 & 2 & 5 \end{matrix})$

Short Answer

Expert verified

The rank of the matrix $(\begin{matrix} 1 & 1 & 2 \\ 2 & 4 & 6 \\ 3 & 2 & 5 \end{matrix})$ is 2 .

Step by step solution

Given information

The given matrix is $(\begin{matrix} 1 & 1 & 2 \\ 2 & 4 & 6 \\ 3 & 2 & 5 \end{matrix})$

Definition of the rank of a matrix

The number of non-zero rows remaining in a row reduced matrix is called the rank of a matrix.

Apply row operations to reduce the matrix into row reduction form

Consider the matrix A = $(\begin{matrix} 1 & 1 & 2 \\ 2 & 4 & 6 \\ 3 & 2 & 5 \end{matrix})$

Multiply row 1 by and subtract the result from row 2.

$A = (\begin{matrix} 1 & 1 & 2 \\ 0 & 2 & 2 \\ 3 & 2 & 5 \end{matrix})$ $R_{2} - 2 R_{1}$

Multiply row 1 by $- 3$ and subtract the result from row 3.

$A = (\begin{matrix} 1 & 1 & 2 \\ 0 & 2 & 2 \\ 0 & - 1 & - 1 \end{matrix})$ role="math" localid="1653389234639" $R_{3} - 3 R_{1}$

Divide the second row by 2.

$A = (\begin{matrix} 1 & 1 & 2 \\ 0 & 1 & 1 \\ 0 & - 1 & - 1 \end{matrix}) \frac{1}{2} R_{2}$

Add row 2 to row 3.

$A = (\begin{matrix} 1 & 1 & 2 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \end{matrix}) R_{3} \pm R_{2}$

The row reduced form of the given matrix is $(\begin{matrix} 1 & 1 & 2 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \end{matrix})$

Here, the total number of non-zero rows is 2. Hence, the rank of the matrix

$A = (\begin{matrix} 1 & 1 & 2 \\ 2 & 4 & 6 \\ 3 & 2 & 5 \end{matrix}) i s 2$

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Find the rank of each of the following matrices.
$(\begin{matrix} 1 & 1 & 2 \\ 2 & 4 & 6 \\ 3 & 2 & 5 \end{matrix})$

Short Answer

Step by step solution

Given information

Definition of the rank of a matrix

Apply row operations to reduce the matrix into row reduction form

One App. One Place for Learning.

Most popular questions from this chapter

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Study anywhere. Anytime. Across all devices.

Find the rank of each of the following matrices.(112246325)

Short Answer

Step by step solution

Given information

Definition of the rank of a matrix

Apply row operations to reduce the matrix into row reduction form

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Geometrical and Physical Optics

Particle Model of Matter

Torque and Rotational Motion

Magnetism and Electromagnetic Induction

Famous Physicists

Force

Study anywhere. Anytime. Across all devices.

Find the rank of each of the following matrices.
$(\begin{matrix} 1 & 1 & 2 \\ 2 & 4 & 6 \\ 3 & 2 & 5 \end{matrix})$