Matrix Algebra and Computational Methods - Module I Quiz
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Questions and Answers

What is the process called when a row in a matrix is replaced by the sum of that row and another row?

Row operation

In a matrix, what is the condition for it to be in row echelon form?

The first non-zero element in each row is the leading entry, and each leading entry is to the right of the leading entry in the previous row.

What is the additional condition for a matrix to be in reduced row echelon form compared to row echelon form?

The leading entry in each row must be 1, and it must be the only non-zero entry in its column.

Define a linear equation in terms of unknowns and constants.

<p>A linear equation in unknowns x, y, z is of the form: ax + by + cz = b, where a, b, c are constants.</p> Signup and view all the answers

What are the coefficients in a linear equation?

<p>The constants a, b, c in the equation ax + by + cz = b are the coefficients of x, y, z.</p> Signup and view all the answers

What is the term used for the constant on the right side of a linear equation?

<p>Constant term</p> Signup and view all the answers

What operation involves changing the order of rows in a matrix?

<p>Interchanging rows</p> Signup and view all the answers

What does a row in a matrix become when it is multiplied by a scalar value?

<p>Scaled</p> Signup and view all the answers

What is the role of the leading entry in each row of a matrix in row echelon form?

<p>It helps determine the structure of the matrix and the position of subsequent leading entries.</p> Signup and view all the answers

What is the key characteristic of the leading entry in each row of a matrix in reduced row echelon form?

<p>The leading entry is always 1.</p> Signup and view all the answers

Flashcards

Row operation

The process of replacing a row in a matrix by the sum of that row and another row.

Row echelon form

A matrix is in row echelon form if the first non-zero element in each row (the leading entry) is to the right of the leading entry in the previous row.

Reduced row echelon form

A matrix is in reduced row echelon form if it is in row echelon form, and the leading entry in each row is 1, and it is the only non-zero entry in its column.

Linear equation

An equation of the form ax + by + cz = d, where a, b, c, and d are constants, and x, y, z are unknowns.

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Coefficients

The constant values multiplying the unknowns in a linear equation (e.g., a, b, c in ax + by + cz = d).

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Constant term

The constant term is the value on the right-hand side of a linear equation. (e.g., d in ax + by + cz = d).

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Interchanging rows

The operation of rearranging the order of rows in a matrix.

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Scaled row

The result of multiplying a row in a matrix by a scalar value.

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Leading entry's role

The leading entry in each row of a matrix in row echelon form determines the structure of the matrix and the position of subsequent leading entries.

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Leading entry in reduced row echelon form

The key characteristic of the leading entry in each row of a matrix in reduced row echelon form is that it is always 1.

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Study Notes

Row Operations

  • Row Replacement: Replacing a row by the sum of itself and a multiple of another row.

Row Echelon Form

  • Leading entry: The first non-zero entry in each row (from left to right).
  • Leading entries staircase: The leading entries of subsequent rows must be to the right of the leading entries of previous rows.
  • Zero rows: All rows consisting entirely of zeroes (if any) must be at the bottom.

Reduced Row Echelon Form

  • Leading entry (1): All leading entries are 1.
  • Zero entries: All entries above each leading entry are 0.

Linear Equation

  • Unknowns: Variables representing unknown values.
  • Constants: Fixed values.
  • Equal sign: Separates the expression with unknowns from the constant.

Linear Equation Terms

  • Coefficients: The numbers multiplying the unknowns in a linear equation.
  • Constant term: The value on the right side of the equal sign in a linear equation.

Matrix Operations

  • Row Swap: Interchanging the positions of two rows in a matrix.
  • Scalar Multiplication: Multiplying a row by a scalar value, resulting in a new row with each element multiplied by the scalar.

Leading Entries

  • Row Echelon Form: Each leading entry is the first non-zero element in a row.
  • Reduced Row Echelon Form: Each leading entry is 1, and all entries above the leading entry in the same column are zero.

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Test your knowledge on Matrices, Linear Equations, Vector Spaces, Matrix Operations, Rank, Determinants, and more from the syllabus of B.Tech 1st Year, II Sem. Department of Mathematics at SR University.

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