Matrix Inverse and Determinants Practice

Front

What is the inverse of a matrix?

Back

The inverse of a matrix A is another matrix, denoted A⁻¹, such that A * A⁻¹ = I, where I is the identity matrix.

Front

How do you find the inverse of a 2x2 matrix?

Back

For a 2x2 matrix A = [[a, b], [c, d]], the inverse A⁻¹ is given by (1/det(A)) * [[d, -b], [-c, a]], where det(A) = ad - bc.

Front

What is a determinant?

Back

The determinant is a scalar value that is a function of the entries of a square matrix, providing important properties such as whether the matrix is invertible.

Front

How do you calculate the determinant of a 2x2 matrix?

Back

For a 2x2 matrix A = [[a, b], [c, d]], the determinant is calculated as det(A) = ad - bc.

Front

What does it mean if the determinant of a matrix is zero?

Back

If the determinant of a matrix is zero, it means the matrix is singular and does not have an inverse.

Front

What is the relationship between a matrix and its inverse?

Back

A matrix and its inverse multiply to give the identity matrix, which has 1s on the diagonal and 0s elsewhere.

Front

How can you determine if two matrices are multiplicative inverses?

Back

Two matrices A and B are multiplicative inverses if A * B = I, where I is the identity matrix.

Front

What is the determinant of the identity matrix?

Back

The determinant of the identity matrix is 1.

Front

How do you find the determinant of a 3x3 matrix?

Back

For a 3x3 matrix, use the rule of Sarrus or cofactor expansion to calculate the determinant.

Front

What is the effect of row operations on the determinant of a matrix?

Back

Row operations affect the determinant: swapping rows changes the sign, multiplying a row by a scalar multiplies the determinant by that scalar, and adding a multiple of one row to another does not change the determinant.