Explain why the determinant of the matrix is equal to zero. ...

Explain why the determinant of the matrix is equal to zero. $$\text { (a) }\left[\begin{array}{rrrr}3 & 4 & -2 & 7 \\1 & 3 & -1 & 2 \\0 & 5 & 7 & 1 \\ 1 & 3 & -1 & 2\end{array}\right]$$ $$\text { (b) }\left[\begin{array}{rrr}3 & 2 & -1 \\-6 & -4 & 2 \\5 & -7 & 9\end{array}\right]$$ $$\text { (c) }\left[\begin{array}{ccc}2 & -4 & 5 \\1 & -2 & 3 \\0 & 0 & 0\end{array}\right]$$ $$\text { (d) }\left[\begin{array}{rrrr}4 & -4 & 5 & 7 \\2 & -2 & 3 & 1 \\4 & -4 & 5 & 7 \\ 6 & 1 & -3 & -3\end{array}\right]$$

Explain why the determinant of the matrix is equal to zero.
$$\text { (a) }\left[\begin{array}{rrrr}3 & 4 & -2 & 7 \\1 & 3 & -1 & 2 \\0 & 5 & 7 & 1 \\
1 & 3 & -1 & 2\end{array}\right]$$
$$\text { (b) }\left[\begin{array}{rrr}3 & 2 & -1 \\-6 & -4 & 2 \\5 & -7 & 9\end{array}\right]$$
$$\text { (c) }\left[\begin{array}{ccc}2 & -4 & 5 \\1 & -2 & 3 \\0 & 0 & 0\end{array}\right]$$
$$\text { (d) }\left[\begin{array}{rrrr}4 & -4 & 5 & 7 \\2 & -2 & 3 & 1 \\4 & -4 & 5 & 7 \\
6 & 1 & -3 & -3\end{array}\right]$$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Matrix Rank and Singularity

The rank of a matrix is the number of linearly independent rows or columns it possesses. If the rank is less than the order of the matrix, the matrix is singular, which is reflected by its determinant being zero. This concept is central in determining the invertibility and the solution spaces of linear systems.

Zero Row or Column

A matrix that contains an entire row or column of zeros has a determinant of zero because the presence of a zero row or column indicates that the corresponding linear transformation compresses the space into a lower dimension, destroying the possibility of an invertible transformation.

Determinant

The determinant is a scalar value associated with a square matrix that provides key information about the matrix, such as whether it is invertible. A nonzero determinant indicates an invertible matrix, while a determinant of zero signifies that the matrix is singular, meaning its rows or columns do not span the full vector space.

Linear Dependence

Linear dependence refers to a situation where at least one row or column in a matrix can be expressed as a linear combination of others. When this occurs, the matrix loses rank and fails to cover the full space, leading directly to a determinant of zero.

Transcript

Please give Ace some feedback