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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.
Explain the concept of the Identity Matrix.
It acts as the '1' in matrix multiplication; multiplying any matrix by the identity matrix returns the original matrix.
Explain the concept of the Inverse Matrix.
It 'undoes' the original matrix. When multiplied together, they result in the Identity Matrix. Only square matrices can have inverses.
Explain the concept of a Determinant.
A scalar value derived from a square matrix that provides information about the matrix's properties, such as invertibility.
Explain the relationship between determinants and invertibility.
A matrix is invertible if and only if its determinant is non-zero. If the determinant is zero, the matrix is singular and does not have an inverse.
Explain the concept of parallel vectors in relation to determinants.
If the determinant of a matrix formed by two vectors is zero, the vectors are parallel, meaning they point in the same or opposite directions.
What are the steps to determine if a 2x2 matrix is invertible?
Calculate the determinant (ad-bc). If the determinant is not zero, the matrix is invertible.
Describe the properties of an identity matrix.
It's a square matrix with 1s on the main diagonal and 0s everywhere else. Multiplying any matrix by it results in the original matrix.
Explain why a zero determinant implies non-invertibility.
A zero determinant indicates that the matrix transforms space in a way that collapses dimensions, making it impossible to 'undo' the transformation with an inverse.
What does the determinant tell us about the area of a parallelogram?
The absolute value of the determinant is equal to the area of the parallelogram defined by the column (or row) vectors of the matrix.
Explain how to find the inverse of a 2x2 matrix.
Swap the diagonal elements, negate the off-diagonal elements, and divide the entire matrix by the determinant.
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.