Finding Inverse Matrices

Which of the following operations is NOT considered an elementary row operation?

What is a key property of all elementary matrices?

According to the Invertible Matrix Theorem, if a square matrix A is invertible, what must be true about the equation $Ax = b$?

If a matrix can be expressed as the product of elementary matrices, what can be concluded about the matrix?

Why is finding the determinant of a matrix useful?

What is the determinant of a 1x1 matrix equal to?

What geometric quantity does the absolute value of the determinant of a 2x2 matrix represent, when the columns of the matrix are considered as vectors?

What are the smaller sub-matrices called when computing determinants by expanding along rows or columns?

In the context of determinants, what is a 'minor'?

What is the relationship between minors and cofactors in determinant calculation?

If you swap two rows of a matrix, how does it affect the determinant?

What does multiplying a row of a matrix by a scalar multiple do to the determinant?

What happens to the determinant if you add a multiple of one row to another row in a matrix?

How is the adjoint of a matrix related to its cofactors?

Which of the following is a condition for a matrix to be invertible, according to the Invertible Matrix Theorem?

What does Cramer's Rule provide?

In Cramer's Rule, if you are solving for $x_i$ in the system $Ax = b$, what do you do to the matrix A?

What characterizes a 'sparse matrix'?

Why might Cramer's Rule be particularly useful, despite potentially high computational cost?

What two operations must be defined for a set of objects to be considered a vector space?

What does it mean for a set to be 'closed' under an operation in the context of vector spaces?

In the vector space $R^3$, what is a general vector written as a linear combination of?

Which of the following is NOT a condition that must be satisfied for a set to be considered a vector space?

Which set can be classified to follow the basic conditions for a Vector Space?

Which of the following sets, with the usual operations of addition and scalar multiplication, forms a vector space?

Given that polynomials with real coefficients follow Vector Space rules, which additional piece of information is needed to prove this?

For any vector space V over a field F, and for any vector v in V, what must be true about the zero vector?

If $c$ is a scalar from the field F and $0$ is the zero vector in vector space V, then what is the result of $c * 0$?

Under what condition, for scalar $c$ in field $F$, and vector $v$ in vector space $V$ can $cv = 0$?

What is a subspace?

Which condition must be met for a subset W of a vector space V to be considered a subspace of V?

What is a key difference between any subset of a vector space and being closed under two separate operations?

What is one of the first things you can check to see if a subset is actually a subspace of another?

What is the null space of a matrix A?

According to Theorem 4.3.12, what statement is correct about the null space?

If a subset $S$ of a vector space $V$ does not contain the zero vector, what can be concluded?

What is a sufficient check to determine if a solution to an equation is a subspace for (or)?

If a matrix A has an inverse, and that inverse is matrix B, what is the result of multiplying matrix A by matrix B?

How does the invertibility of an elementary matrix relate to its row operations?

According to the Invertible Matrix Theorem, which of the following conditions ensures that a square matrix A is invertible?

Why is it useful to know if a matrix can be expressed as the product of elementary matrices?

If the columns of a 2x2 matrix are viewed as vectors forming a parallelogram, and the determinant of this matrix is zero, what can be said about the area of the parallelogram?

In calculating the determinant using cofactor expansion, how do you determine the sign (positive or negative) associated with each cofactor?

How does the determinant of a matrix $A$ relate to the determinant of its adjoint, adj$(A)$, when $A$ is a $n \times n$ matrix?

According to the Invertible Matrix Theorem, which of the following statements is true about the determinant of an invertible matrix A?

In using Cramer's Rule to solve a system of linear equations, what happens if the determinant of the coefficient matrix A is zero?

What is the primary advantage of using cofactor expansion to find the determinant of a sparse matrix?

If a set of vectors in $R^n$ fails to meet the condition of closure under scalar multiplication, what can be concluded about the set?

In the context of vector spaces, what does it mean for a vector space to be 'closed' under vector addition?

Consider a set $S$ of all 2x2 matrices with real entries where the determinant of each matrix is equal to 1. With standard matrix addition, and scalar multiplication, is this set closed under addition?

What distinguishes a subspace from any arbitrary subset of a vector space?

If W is a subspace of V, and V is a vector space, which of the following operations when applied to elements of W is guaranteed to result in an element that is also in W?

What condition regarding the zero vector is necessary for a subset $W$ of a vector space $V$ to potentially be a subspace of $V$?

If $W$ is the null space of matrix $A$, then for any vector $x$ in $W$, what is the result of $Ax$?

Theorems regarding the Null Space detail that the set of solutions to $Ax = 0$ form a...

If a non-empty subset $S$ of a vector space $V$ does NOT satisfy the condition that for all vectors $u$ and $v$ in $S$, the sum $u + v$ is also in $S$, what can be definitively concluded?

To determine if the solution set of a nonhomogeneous linear system $Ax = b$ (where $b \neq 0$) is a subspace, what is the most efficient first step?