Finding Determinant & Inverse of Matrices Flashcard

Front

What is the determinant of a 2x2 matrix?

Back

For a 2x2 matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \), the determinant is calculated as \( det(A) = ad - bc \).

Front

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

Back

For a 2x2 matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \), the inverse is given by \( A^{-1} = \frac{1}{det(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \) if \( det(A) \neq 0 \).

Front

Calculate the determinant of the matrix \( \begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix} \).

Back

The determinant is \( det = (3)(4) - (2)(1) = 12 - 2 = 10 \).

Front

What is the determinant of the matrix \( \begin{pmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{pmatrix} \)?

Back

The determinant is calculated using cofactor expansion or row reduction. The answer is -24.

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

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

Back

For a 3x3 matrix \( A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \), the determinant is \( det(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \).

Front

What is the formula for the determinant of a 2x2 matrix?

Back

For a 2x2 matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \), the formula is \( det(A) = ad - bc \).

Front

Find the inverse of the matrix \( \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix} \).

Back

The inverse is \( \begin{pmatrix} 4/5 & -3/5 \\ -1/5 & 2/5 \end{pmatrix} \).

Front

What is the determinant of the matrix \( \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \)?

Back

The determinant is \( det = (0)(0) - (1)(1) = -1 \).

Front

How do you determine if a matrix is invertible?

Back

A matrix is invertible if its determinant is non-zero.