Properties and Inverses of Diagonal Matrices

This video tutorial covers diagonal matrices, starting with their definition and properties. It explains how to determine if a diagonal matrix is singular or invertible and how to find the inverse of an invertible diagonal matrix. The video also demonstrates how to raise a diagonal matrix to a power by raising each diagonal element to the power. Examples are provided to illustrate these concepts, and the video concludes with a summary of the key points discussed.

What is a defining characteristic of a diagonal matrix?

• All entries are equal.

• All diagonal entries are zero.

• All non-diagonal entries are zero.

• All entries are non-zero.

When is a diagonal matrix considered singular?

• When any diagonal element is zero.

• When it has an inverse.

• When all diagonal elements are non-zero.

• When all non-diagonal elements are zero.

How can you determine if a diagonal matrix is invertible?

• By checking if all diagonal elements are non-zero.

• By checking if all non-diagonal elements are non-zero.

• By ensuring the matrix is square.

• By calculating the determinant.

What is the inverse of a diagonal matrix with non-zero diagonal elements?

• A matrix with the same diagonal elements.

• A matrix with reciprocal diagonal elements.

• A matrix with negative diagonal elements.

• A matrix with zero diagonal elements.

If a diagonal matrix has a diagonal element of 3, what is the corresponding element in its inverse?

• -3

• 3

• 1/3

• 0

What happens to the non-diagonal elements when finding the inverse of a diagonal matrix?

• They become negative.

• They remain zero.

• They become non-zero.

• They become reciprocal.

How do you raise a diagonal matrix to a power?

• Add the power to each diagonal element.

• Multiply all elements by the power.

• Raise each diagonal element to the power.

• Subtract the power from each diagonal element.

What is the result of raising a diagonal element of -2 to the third power?

• 6

• -8

• 8

• -6

If a diagonal matrix has a diagonal element of 4, what is the result when it is raised to the third power?

• 8

• 64

• 12

• 16

What remains unchanged when a diagonal matrix is raised to a power?

• Non-diagonal elements

• Matrix size

• Matrix determinant

• Diagonal elements