Chapter 6 - Matrices

What is a square matrix?

A matrix with the same number of rows and columns.

What is a zero matrix?

A matrix in which all entries are zero.

How is the zero matrix denoted?

By 0.

What is an identity matrix?

A square matrix with 1s on the leading diagonal and 0s elsewhere.

How are identity matrices denoted?

By Ik where k is the size of the matrix.

What is the 3×3 identity matrix?

A matrix with 1s on the main diagonal and 0s elsewhere.

How do you add or subtract matrices?

By adding or subtracting corresponding elements.

When can matrices be added or subtracted?

Only when they are the same size.

How do you multiply a matrix by a scalar?

Multiply every element in the matrix by the scalar.

When can two matrices be multiplied?

When the number of columns in the first equals the number of rows in the second.

What does multiplicatively conformable mean?

Matrices that can be multiplied together.

How do you multiply two matrices?

Multiply rows by columns and add the products.

What is the determinant of a 2×2 matrix?

ad − bc.

When is a matrix singular?

When its determinant is 0.

When is a matrix non-singular?

When its determinant is not 0.

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

By reducing it to 2×2 determinants using expansion.

What is a minor of an element?

The determinant of the 2×2 matrix left after removing its row and column.

What is the inverse of a matrix?

The matrix M⁻¹ such that MM⁻¹ = M⁻¹M = I.

What is the inverse of a 2×2 matrix?

1/det(M) × [[d, −b], [−c, a]].

If A and B are non-singular, what is (AB)⁻¹?

B⁻¹A⁻¹.

What is the transpose of a matrix?

A matrix formed by interchanging rows and columns.

What is the first step in finding the inverse of a 3×3 matrix?

Find the determinant of the matrix.

What is the second step in finding the inverse of a 3×3 matrix?

Form the matrix of minors.

What is the third step in finding the inverse of a 3×3 matrix?

Form the matrix of cofactors using alternating signs.

What is the fourth step in finding the inverse of a 3×3 matrix?

Take the transpose of the matrix of cofactors.

What is the fifth step in finding the inverse of a 3×3 matrix?

Multiply the transpose of cofactors by 1/det(A).

What is the formula for the inverse of a 3×3 matrix?

A⁻¹ = 1/det(A) × Cᵀ.

How can a system Ax = v be solved?

By x = A⁻¹v.

When is a system of linear equations consistent?

When it has at least one solution.

When is a system of linear equations inconsistent?

When it has no solution.