2- Special matrices

What is the transpose of any matrix?

• The transpose of any matrix is one in which the rows and columns are interchanged

• The first row becomes the first column etc

What are the steps to find the inverse of a 3 x 3 matrix?

• Compute the Determinant of the matrix A.

• Find the Matrix of Cofactors.

• Transpose the Matrix of Cofactors to get the Adjugate matrix.

• Multiply the Adjugate by 1/det(A) to obtain the inverse.

What is the transpose of matrix A denoted as?

A^T

When is a square matrix symmetric?

What is the rule for transposing the product of two matrices?

What is a symmetric matrix?

A square matrix A is symmetric if A^T=A

How can you recognize a symmetric matrix?

The elements are mirrored across the leading diagonal (top-left to bottom-right).

What is an antisymmetric (or skew-symmetric) matrix?

A square matrix A is antisymmetric if A^T=−A

What must be true about the leading diagonal of an antisymmetric matrix?

All elements on the leading diagonal must be zero.

What happens when you add a square matrix to its transpose?

A+A^T is always symmetric.

What happens when you subtract a square matrix from its transpose?

A-A^T is always anti-symmetric.

What is a diagonal matrix?

A square matrix where all elements off the leading diagonal are zero.

What condition must be true for a matrix A to be diagonal?

a_ij=0 for all i≠j (only diagonal elements can be non-zero).

What is an identity matrix?

A diagonal matrix where all diagonal elements are 1.

What is another name for the identity matrix?

It is also called the unit matrix.

What is the key property of an identity matrix when multiplied by another matrix?

For any matrix A, multiplying by I (Identity matrix) gives the same matrix A

What is the definition of the inverse of a square matrix A?

• A matrix M is the inverse of A if AM=MA=I.

• We write M=A⁻¹

What happens when you multiply a matrix by its inverse?

The result is the identity matrix:

If M is the inverse of A, what is the inverse of M?

If AB=C, how do you solve for A?

Multiply by B⁻¹ on the right:

What is the difference between pre-multiplying and post-multiplying?

How can a system of simultaneous equations be written in matrix form?

How do you solve for X in the matrix equation AX=H?

Multiply both sides by A⁻¹

What is the formula for the inverse of a 2×2 matrix?

What is the determinant of a 2×2 matrix?

When does a 2×2 matrix not have an inverse?

If the determinant is zero (ad−bc=0), the matrix has no inverse and is called singular.

What is the first step in finding the inverse of a 2×2 matrix?

Ensure that the determinant is not zero