Question

Explain why the determinant of each matrix is equal to zero. (a) $\left[\begin{array}{rrr}2 & -4 & 5 \\ 1 & -2 & 3 \\ 0 & 0 & 0\end{array}\right]$ (b) $\left[\begin{array}{rrrr}4 & -4 & 5 & 7 \\ 2 & -2 & 3 & 1 \\ 4 & -4 & 5 & 7 \\ 6 & 1 & -3 & -3\end{array}\right]$

Name: Problem 102, Chapter 8: Precalculus
Uploaded: 2021-02-12T05:45:06-08:00
Duration: 1 min 37 s
Channel: Anthony Ramos
Description: Explain why the determinant of each matrix is equal to zero. (a) [ 2 -4 5 1 -2 3 0 0 0 ] (b) [ 4 -4 5 7 2 -2 3 1 4 -4 5 7 6 1 -3 -3 ]

   Explain why the determinant of each matrix is equal to zero.
(a) $\left[\begin{array}{rrr}2 & -4 & 5 \\ 1 & -2 & 3 \\ 0 & 0 & 0\end{array}\right]$
(b) $\left[\begin{array}{rrrr}4 & -4 & 5 & 7 \\ 2 & -2 & 3 & 1 \\ 4 & -4 & 5 & 7 \\ 6 & 1 & -3 & -3\end{array}\right]$

Precalculus

Ron Larson 10th Edition

Chapter 8, Problem 102

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Solved by Expert Anthony Ramos

02/12/2021

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The determinant of a matrix is a special number that can be calculated from a square matrix. It is used to find the inverse of a matrix, to solve systems of equations, and many other things. Show more…

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Explain why the determinant of each matrix is equal to zero. (a) $\left[\begin{array}{rrr}2 & -4 & 5 \\ 1 & -2 & 3 \\ 0 & 0 & 0\end{array}\right]$ (b) $\left[\begin{array}{rrrr}4 & -4 & 5 & 7 \\ 2 & -2 & 3 & 1 \\ 4 & -4 & 5 & 7 \\ 6 & 1 & -3 & -3\end{array}\right]$

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Key Concepts

Zero or Identical Rows

The occurrence of a zero row or identical rows in a matrix is a specific case of linear dependence. A zero row contributes nothing to the matrix’s rank, while identical rows mean that the information is redundant. In both cases, the presence of such rows guarantees that the determinant is zero, as the matrix does not have full rank.

Linear Dependence

Linear dependence refers to the condition where at least one row or column of a matrix can be expressed as a linear combination of the others. This dependence implies a loss of dimensionality in the space spanned by the rows or columns, which is directly linked to having a determinant of zero.

Determinant

The determinant is a scalar value that provides important information about a matrix, such as whether it is invertible and the volume scaling effect of the linear transformation it represents. When the determinant is zero, it indicates that the matrix is singular and does not have an inverse.

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