AP PreCalculus Flashcards: The Inverse and Determinant of a Matrix
Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026
Review key ideas with interactive flashcards. This set includes 11 cards to help you master important concepts.
What is the term for a square matrix that has an inverse?
A square matrix that has an inverse is called an invertible or non-singular matrix.
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What is the term for a square matrix that has an inverse?
A square matrix that has an inverse is called an invertible or non-singular matrix.
Given $A \\cdot A^{-1} = I$, what is the role of $I$ in this equation?
$I$ is the identity matrix, which serves as the multiplicative identity in matrix algebra, similar to the number 1 in scalar multiplication.
A square matrix has a determinant of 1. Is this matrix invertible?
Yes, the matrix is invertible because its determinant is not zero.
What is the identity matrix, $I$?
The identity matrix is a square matrix consisting of 1s on the diagonal from the top left to bottom right and 0s everywhere else.
How is the determinant of a $2 \\times 2$ matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ calculated?
The determinant is found using the formula $\det(A) = ad - bc$.
If the determinant of a matrix is 0, what can you conclude about its invertibility?
If a matrix's determinant is 0, the matrix is not invertible, meaning it does not have an inverse.
What is the relationship between a matrix's determinant and its invertibility?
A matrix is invertible if its determinant is non-zero, and it is non-invertible (or singular) if its determinant is zero.
What is the key condition for a square matrix to have an inverse?
A square matrix $A$ has an inverse if and only if its determinant is not equal to zero ($\det(A) \ne 0$).
How does the determinant's value relate to the vectors forming the matrix's columns?
A non-zero determinant indicates the column vectors are linearly independent, while a zero determinant indicates they are linearly dependent.
What is the result of multiplying a square matrix by its inverse?
The product of a square matrix and its inverse, when it exists, is the identity matrix of the same size.
Calculate the determinant of the matrix $A = \begin{pmatrix} 4 & 1 \\ 2 & 3 \end{pmatrix}$.
The determinant is $(4)(3) - (1)(2) = 12 - 2 = 10$.