Question

Find the rank of each of the following matrices from its echelon matrix, and comment on the question of nonsingularity. \((a) A=\left[\begin{array}{rrr}1 & 5 & 1 \\ 0 & 3 & 9 \\ -1 & 0 & 0\end{array}\right]\) (b) \(B=\left[\begin{array}{rrr}0 & -1 & -4 \\ 3 & 1 & 2 \\ 6 & 1 & 0\end{array}\right]\) \((c) \subset=\left[\begin{array}{llll}7 & 6 & 3 & 3 \\ 0 & 1 & 2 & 1 \\ 8 & 0 & 0 & 8\end{array}\right]\) (d) \(D=\left[\begin{array}{lllr}2 & 7 & 9 & -1 \\ 1 & 1 & 0 & 1 \\ 0 & 5 & 9 & -3\end{array}\right]\)

Short answer

(a) Rank 3, nonsingular; (b) Rank 3, nonsingular; (c) Rank 3, nonsingularity not applicable; (d) Rank 2, nonsingularity not applicable.

Step-by-step solution

Matrix A - Reduce to Echelon Form

Start with matrix \( A \):\[ A = \begin{bmatrix} 1 & 5 & 1 \ 0 & 3 & 9 \ -1 & 0 & 0 \end{bmatrix} \]Use row operations to convert it into echelon form. Step 1: Add row 1 to row 3 to get a new row 3:\[ \begin{bmatrix} 1 & 5 & 1 \ 0 & 3 & 9 \ 0 & 5 & 1 \end{bmatrix} \]The matrix is in echelon form with three non-zero rows.

Matrix A - Determine Rank

The rank of a matrix is the number of non-zero rows in its echelon form. Here, matrix A has three non-zero rows, so the rank of \( A \) is 3.

Matrix A - Nonsingularity

A square matrix is nonsingular if its rank equals the number of rows (or columns). Since \( A \) is a 3x3 matrix and its rank is 3, it is nonsingular.

Matrix B - Reduce to Echelon Form

Start with matrix \( B \):\[ B = \begin{bmatrix} 0 & -1 & -4 \ 3 & 1 & 2 \ 6 & 1 & 0 \end{bmatrix} \]Step 1: Swap row 1 and row 2 to move the leading 1 into position:\[ \begin{bmatrix} 3 & 1 & 2 \ 0 & -1 & -4 \ 6 & 1 & 0 \end{bmatrix} \]Step 2: Eliminate the entry below the leading 1 in the first column:Subtract 2 times row 1 from row 3:\[ \begin{bmatrix} 3 & 1 & 2 \ 0 & -1 & -4 \ 0 & -1 & -4 \end{bmatrix} \]Step 3: Add row 2 to row 3 to eliminate the -1 in row 3:\[ \begin{bmatrix} 3 & 1 & 2 \ 0 & -1 & -4 \ 0 & 0 & -8 \end{bmatrix} \]The matrix is in echelon form with three non-zero rows.

Matrix B - Determine Rank

The rank of B is the number of non-zero rows in its echelon form. Matrix B has three non-zero rows, so the rank of \( B \) is 3.

Matrix B - Nonsingularity

Since B is a 3x3 matrix and the rank is 3, B is nonsingular.

Matrix C - Reduce to Echelon Form

Start with matrix \( C \):\[ C = \begin{bmatrix} 7 & 6 & 3 & 3 \ 0 & 1 & 2 & 1 \ 8 & 0 & 0 & 8 \end{bmatrix} \]Step 1: Subtract row 1 from row 3 to eliminate the 8 in row 3:\[ \begin{bmatrix} 7 & 6 & 3 & 3 \ 0 & 1 & 2 & 1 \ 1 & -6 & -3 & 5 \end{bmatrix} \]Step 2: Subtract 1/7 times row 1 from row 3:\[ \begin{bmatrix} 7 & 6 & 3 & 3 \ 0 & 1 & 2 & 1 \ 0 & -6.86 & -3.43 & 4.57 \end{bmatrix} \]Step 3: Add row 2 times 6.86 to row 3:\[ \begin{bmatrix} 7 & 6 & 3 & 3 \ 0 & 1 & 2 & 1 \ 0 & 0 & 10 & 11 \end{bmatrix} \]The matrix is in echelon form with three non-zero rows.

Matrix C - Determine Rank

The rank of C is the number of non-zero rows in its echelon form. Matrix C has three non-zero rows, so the rank of \( C \) is 3.

Matrix C - Nonsingularity

Since C is a 3x4 matrix and its rank is 3, while it is full row rank, it is not square, so non-singularity does not apply.

Matrix D - Reduce to Echelon Form

Start with matrix \( D \):\[ D = \begin{bmatrix} 2 & 7 & 9 & -1 \ 1 & 1 & 0 & 1 \ 0 & 5 & 9 & -3 \end{bmatrix} \]Step 1: Subtract 0.5 times row 1 from row 2:\[ \begin{bmatrix} 2 & 7 & 9 & -1 \ 0 & -2.5 & -4.5 & 1.5 \ 0 & 5 & 9 & -3 \end{bmatrix} \]Step 2: Add 2 times row 2 to row 3:\[ \begin{bmatrix} 2 & 7 & 9 & -1 \ 0 & -2.5 & -4.5 & 1.5 \ 0 & 0 & 0 & 0 \end{bmatrix} \]The matrix is in echelon form with two non-zero rows.

Matrix D - Determine Rank

The rank of D is the number of non-zero rows in its echelon form. Matrix D has two non-zero rows, so the rank of \( D \) is 2.

Matrix D - Nonsingularity

Since D is a 3x4 matrix, it's non-square, but it does not have full rank equal to the number of columns or rows for a square subset, so non-singularity does not apply.

Key concepts

Echelon Form

Echelon form is an incredibly useful concept when working with matrices. This form makes it much easier to identify the rank and other properties of a matrix. To convert a matrix to its echelon form, follow these guidelines:

• The matrix will have zeros below each leading coefficient in each successive row.
• Each leading coefficient is to the right of the ones above it.
• Any rows that contain only zeros should be at the bottom of the matrix.

Transforming a matrix into this form is achieved through row operations, which we will discuss in more detail later. For example, after performing these operations on matrix A from the exercise, it changes to:
\[\begin{bmatrix}1 & 5 & 1 \0 & 3 & 9 \0 & 5 & 1\end{bmatrix}\]This simple structure helps in quickly determining important properties like rank; here, matrix A has three non-zero rows, indicating its rank is 3.

Nonsingularity

The concept of nonsingularity relates closely to the structure and properties of matrices in linear algebra. A square matrix is considered nonsingular if it meets certain criteria:

• The rank equals the number of rows (or columns).
• There exists a unique solution for a set of linear equations represented by the matrix.
• Its determinant is non-zero, indicating the matrix is invertible.

For the matrices characterized in the exercise, we can check their nonsingularity after identifying their ranks. For example, matrix A is a square matrix (3x3) and has a rank of 3. This makes it nonsingular, capable of being inverted, and every system of equations it represents has a unique solution. On the other hand, matrix C is a 3x4 matrix, making it non-square, so nonsingularity isn't applicable. Understanding these characteristics ensures clarity about the potential solutions of systems of equations represented by these matrices.

Row Operations

Row operations are the building blocks of converting matrices to a simpler structure, such as the echelon form. These operations are fundamental tools in linear algebra, and they are categorized in three primary types:

• Row multiplication: Multiplying all elements of a row by a non-zero scalar.
• Row swapping: Exchanging the positions of two rows within the matrix.
• Row addition: Adding or subtracting multiples of one row to another row to form a new row.

Applying these operations methodically will lead any given matrix towards its echelon form efficiently. For instance, in the given exercise, we perform row addition on matrix A, adding row 1 to row 3 to eliminate −1 in the first row position of the third row. This process simplifies the structure of matrices, making it easier to analyze and determine their rank, solution possibilities, and other attributes. By mastering row operations, one gains a powerful toolkit for manipulating and understanding matrices on a deeper level.