For any matrix to be in row reduced form which of the following conditions is not satisfied?
zero row is at the top of the matrix
Understanding Row Reduced Form of a Matrix
A matrix is said to be in row reduced form (also known as reduced row echelon form) if it satisfies a specific set of conditions. These conditions make the matrix unique for a given original matrix and are the result of applying elementary row operations through a process like Gaussian-Jordan elimination. Understanding these conditions is crucial in linear algebra, especially for solving systems of linear equations or finding the rank of a matrix.
Conditions for Row Reduced Form
For a matrix to be in row reduced form, the following properties must hold:
- Any row consisting entirely of zeros must be at the bottom of the matrix.
- For each non-zero row, the first non-zero entry from the left (called the leading entry or pivot) is a 1. This is also called a leading one.
- For any two successive non-zero rows, the leading one in the lower row appears to the right of the leading one in the row immediately above it.
- Each column that contains a leading one has zeros in every other position (above and below the leading one).
Analyzing the Given Conditions
Let's examine each condition provided in the options against the standard conditions for row reduced form:
- Each leading one is to the right of one preceding row: This condition aligns with the third standard condition, stating that the leading one in a lower row must be to the right of the leading one in the row above it. This condition is satisfied in a row reduced matrix.
- Any row if has zero is at the bottom: This condition is slightly awkwardly phrased but likely refers to rows that are entirely zero (zero rows). The first standard condition explicitly states that all zero rows must be at the bottom of the matrix. This condition is satisfied in a row reduced matrix.
- zero row is at the top of the matrix: This condition directly contradicts the first standard condition which requires all zero rows to be at the bottom of the matrix. Therefore, this condition is not satisfied in a row reduced matrix.
- any leading entry is one: This condition aligns with the second standard condition, requiring that the first non-zero entry (leading entry) in any non-zero row must be a 1 (a leading one). This condition is satisfied in a row reduced matrix.
Identifying the Condition Not Satisfied
Based on the analysis, the condition that is not satisfied for a matrix to be in row reduced form is the statement that a zero row is at the top of the matrix. In fact, zero rows must always be at the bottom in row reduced form.
| Condition from Option | Relation to RREF Standard Conditions | Is this condition satisfied in RREF? |
|---|---|---|
| Each leading one is to the right of one preceding row | Matches standard condition 3 | Yes |
| Any row if has zero is at the bottom | Matches standard condition 1 (for zero rows) | Yes |
| zero row is at the top of the matrix | Contradicts standard condition 1 | No |
| any leading entry is one | Matches standard condition 2 | Yes |
Revision Table: Row Reduced Form Key Concepts
| Property | Description |
|---|---|
| Leading Entry | The first non-zero element in a row from the left. |
| Leading One (Pivot) | The leading entry must be 1. |
| Zero Rows | Rows containing only zeros must be at the bottom. |
| Leading One Position | Leading one in a lower row is to the right of the leading one in the row above. |
| Pivot Column | Columns containing a leading one have zeros elsewhere. |
Additional Information: Importance of Row Reduced Form
The row reduced form of a matrix is unique. This means that no matter which sequence of elementary row operations you use, you will always arrive at the same row reduced form for a given matrix. This uniqueness makes RREF a powerful tool for various tasks in linear algebra, including:
- Solving systems of linear equations (the solution can often be read directly from the augmented matrix in RREF).
- Finding the rank of a matrix (the number of non-zero rows in RREF equals the rank).
- Determining the invertibility of a square matrix (a square matrix is invertible if and only if its RREF is the identity matrix).
- Finding a basis for the row space, column space, and null space of a matrix.
The process of transforming a matrix into row reduced form involves applying elementary row operations: swapping two rows, multiplying a row by a non-zero scalar, and adding a multiple of one row to another row. This process is commonly known as Gaussian-Jordan elimination.
Important Questions from Matrix Algebra
- If A = \( \left[\begin{array}{cc}0 & 1 \\ −1 & 0\end{array}\right]\) and (aI 2 + bA)2 = A, then
If A = \(\left[\begin{array}{cc}2 & −3 \\3 & 5\end{array}\right]\), then which of the following statements are correct?
A. A is a square matrix
B. A−1 exists
C. A is a symmetric matrix
D. |A| = 19
E. A is a null matrix
Choose the correct answer from the options given below.
If A is Square Matrix of order 3, then product of A and its transpose is
What is the transformation matrix M that transforms a square in the xy-plane defined by (1, 1) T, (-1, 1) T, (-1, -1) T and (1, -1) T to a parallelogram whose corresponding vertices are (2, 1) T, (0, 1) T, (-2, -1) T and (0, -1) T?
Let \(A = \left[ {\begin{array}{*{20}{c}} 1&1&0\\ 0&1&0\\ 1&1&0\\ 0&0&1 \end{array}} \right]\) and \(B = \left[ {\begin{array}{*{20}{c}} 1&0&0&0\\ 0&1&1&0\\ 1&0&1&1\\ \end{array}} \right]\) Find the boolean product A ⊙ B of the two matrices.
