All Flashcards
How do you find the inverse of a 2x2 matrix?
Calculate the determinant (ad-bc), swap a and d, negate b and c, divide each element by the determinant.
How do you determine if two vectors are parallel using determinants?
Form a matrix with the vectors as columns (or rows), calculate the determinant. If the determinant is zero, the vectors are parallel.
How do you calculate the determinant of a 2x2 matrix?
Multiply the elements on the main diagonal (ad), multiply the elements on the other diagonal (bc), subtract the second product from the first (ad - bc).
How do you check if a matrix has an inverse?
Calculate the determinant. If the determinant is not equal to zero, the matrix has an inverse.
Given two vectors, how do you find the area of the parallelogram they span?
Create a 2x2 matrix using the vectors as columns, find the determinant, take the absolute value.
What is the first step to finding the inverse of a matrix?
Calculate the determinant of the matrix.
How do you determine if a matrix is singular?
Calculate its determinant. If the determinant is zero, the matrix is singular.
How to solve for x if det(A) = 0, where A is a matrix with 'x' as an element?
Calculate the determinant in terms of 'x', set the expression equal to zero, solve the resulting equation for 'x'.
How to find the determinant of a 3x3 matrix without a calculator?
Use cofactor expansion along a row or column, multiplying each element by its cofactor and summing the results. (Use calculator on exam)
How do you use the inverse of a matrix to solve a system of linear equations?
Represent the system as AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix. Then, X = A⁻¹B.
What are the differences between a matrix and its inverse?
Matrix: transforms a vector | Inverse: 'undoes' the transformation of the original matrix.
What are the differences between finding the inverse of a 2x2 matrix vs a 3x3 matrix?
2x2: Use the formula (swap diagonal, negate off-diagonal, divide by determinant) | 3x3: Use calculator or cofactor expansion (more complex).
What are the differences between invertible and non-invertible matrices?
Invertible: det(A) ≠ 0, has an inverse | Non-Invertible: det(A) = 0, does not have an inverse.
What are the differences between the identity matrix and a zero matrix?
Identity Matrix: 1s on main diagonal, 0s elsewhere, acts as '1' in multiplication | Zero Matrix: all elements are 0, acts as '0' in addition/multiplication.
What are the differences between parallel and perpendicular vectors in terms of determinants?
Parallel: determinant of matrix formed by vectors is 0 | Perpendicular: determinant provides no direct information (dot product is 0).
What are the differences between a matrix and a determinant?
Matrix: a rectangular array of numbers | Determinant: a scalar value calculated from a square matrix.
What is an Identity Matrix?
A square matrix with 1s on the main diagonal and 0s everywhere else.
What is an Inverse Matrix?
A matrix that, when multiplied by the original matrix, results in the identity matrix.
What is a Determinant?
A scalar value calculated from a square matrix that indicates its invertibility.
What does it mean for a matrix to be invertible?
A matrix is invertible if it has an inverse, meaning its determinant is not zero.
What is a singular matrix?
A matrix that does not have an inverse because its determinant is zero.
What is the main diagonal of a matrix?
The diagonal from the top left to the bottom right of a square matrix.
What are parallel vectors in the context of determinants?
Vectors that point in the same or opposite directions, indicated by a determinant of zero.
What is the size of an identity matrix?
Always n x n (square), where n is the number of rows and columns.
What is the relationship between the determinant and area?
The absolute value of the determinant of a 2x2 matrix formed by two column vectors is the area of the parallelogram they span.
Define 'invertibility condition'.
A square matrix A has an inverse only if its determinant is NOT zero: det(A) ≠ 0.