Question

Find the rank of the matrix A where: (i) \(\quad \begin{array}{rrrrrr} & \mid 1 & 3 & 1 & -2 & -3 \mid \\ & A=\mid 1 & 4 & 3 & -1 & -4 \mid \\ & \mid 2 & 3 & -4 & -7 & -3 \\ & \mid 3 & 8 & 1 & -7 & -8 \mid\end{array}\) (ii) \(\quad \begin{aligned}&\mid 1 & 2 & -3 \mid \\\&A= & \mid 2 & 1 & 0 \mid \\\&\mid-2 & -1 & 3 \\\&\mid-1 & 4 & -2\end{aligned} \mid\) (iii) \(\quad \begin{array}{rr}\mid 1 & 3 \mid \\ A=\mid & -2 \mid \\ \mid 5 & -1 \mid \\ \mid-2 & 3\end{array} \mid\)

Short answer

The rank of matrix A in each case is: (i) Rank(A) = 3 (ii) Rank(A) = 2 (iii) Rank(A) = 1

Step-by-step solution

Row Operations

Perform the following row operations: 1. \(R_2 = R_2 - R_1\) 2. \(R_3 = R_3 - 2R_1\) 3. \(R_4 = R_4 - 3R_1\) 4. \(R_4 = R_4 - \frac{1}{2}R_3\) After performing these operations, we get: \(A = \begin{bmatrix} 1 & 3 & 1 & -2 & -3\\ 0 & 1 & 2 & 1 & -1\\ 0 & 0 & -6 & -3 & 3\\ 0 & 0 & 0 & 0 & 0\end{bmatrix}\) Step 2: Count the number of non-zero rows

Non-Zero Rows

There are 3 non-zero rows in the row echelon form of A. Hence, the rank of matrix A is 3. (ii) Let's find the rank of the matrix A: \( A = \begin{bmatrix} 1 & 2 & -3\\ 2 & 1 & 0\\ -2 & -1 & 3\\ -1 & 4 & -2\end{bmatrix}\) Step 1: Convert the matrix to row echelon form

Row Operations

Perform the following row operations: 1. \(R_2 = R_2 - 2R_1\) 2. \(R_3 = R_3 + 2R_1\) 3. \(R_4 = R_4 + R_1\) 4. \(R_4 = R_4 - 3R_2\) After performing these operations, we get: \(A = \begin{bmatrix} 1 & 2 & -3\\ 0 & -3 & 6\\ 0 & 0 & 0\\ 0 & 0 & 0\end{bmatrix}\) Step 2: Count the number of non-zero rows

Non-Zero Rows

There are 2 non-zero rows in the row echelon form of A. Hence, the rank of matrix A is 2. (iii) Let's find the rank of the matrix A: \(A = \begin{bmatrix} 1 & 3\\ -2 & -6\\ 5 & -1\\ -2 & 3\end{bmatrix}\) Step 1: Convert the matrix to row echelon form

Row Operations

Perform the following row operations: 1. \(R_2 = R_2 + 2R_1\) 2. \(R_3 = R_3 - 5R_1\) 3. \(R_4 = R_4 + 2R_1\) After performing these operations, we get: \(A = \begin{bmatrix} 1 & 3\\ 0 & 0\\ 0 & 0\\ 0 & 0\end{bmatrix}\) Step 2: Count the number of non-zero rows

Non-Zero Rows

There is 1 non-zero row in the row echelon form of A. Hence, the rank of matrix A is 1.

Key concepts

Row Operations

Row operations are incredibly useful for simplifying matrices. They involve changing a matrix from one form to another by manipulating its rows. Here are the types of operations you can typically perform on matrix rows:

• Row Swapping: Changing the positions of two rows. This does not affect the determinant or rank of the matrix.
• Row Multiplication: Multiplying all elements of a row by a non-zero scalar. While this changes the determinant, it keeps the same linear relationships in the rows.
• Row Addition/Subtraction: Adding or subtracting one row from another. This is a strategic operation to create zeros and simplify the matrix.

Let's consider an example, if you have a matrix and want to make the first value of the second row zero, you can subtract (or add, depending on the scenario) a multiple of the first row to the second row. These operations are essential for transforming matrices into easier forms such as the Row Echelon Form (REF) while maintaining their rank.

Row Echelon Form

The row echelon form (REF) of a matrix is a type of matrix simplification that makes it easier to understand and solve linear equations. When a matrix is in row echelon form:

• All non-zero rows are above any rows of all zeros.
• The leading entry of each non-zero row after the first occurs to the right of the leading entry of the previous row.
• The leading entry in any non-zero row is 1 (although not strictly necessary for simpler echelon forms, it is often preferred).

Achieving REF is usually done by performing row operations until each row's leading coefficient (also known as a pivot) is set, and non-essential numbers are zeroed out. Having a matrix in this form is particularly helpful when counting the number of non-zero rows, which correspond to the rank of the matrix. It helps in finding solutions to systems of linear equations and determining the span of the column vectors.

Non-Zero Rows

Non-zero rows are crucial when determining the rank of a matrix. In the context of row echelon form, a non-zero row is any row that contains at least one element that is not zero. Here's how they contribute to understanding matrices:

• Rank: The rank of a matrix is defined as the maximum number of linearly independent row vectors in the matrix. After transforming a matrix to its row echelon form, the rank is simply the number of non-zero rows. This gives an easy way to assess the dimensions of the row space.
• Simplification: Non-zero rows represent important linear combinations, while rows that are reduced to all zeros do not contribute any new information.

In performing exercises that involve finding the rank, counting non-zero rows becomes very straightforward once the matrix is simplified to row echelon form. This makes understanding and solving linear system problems more approachable and less intimidating.