Row Operations
Name the three types of elementary row operations.
Interchange two rows, multiply a row by a non-zero constant, add a multiple of one row to another
What is the formula for the determinant of a 2×2 matrix
The determinant is given by det(A)=ad−bc.
What is the defining equation that relates an eigenvector v and its corresponding eigenvalue λ?
The defining equation is Av=λv, where A is a square matrix, v is a nonzero vector, and λ is a scalar.
What property must be preserved when performing row operations on an augmented matrix?
The solution set of the system must remain unchanged.
What happens to the determinant of a matrix if one of its rows is multiplied by a scalar k?
The determinant is also multiplied by k. For example, multiplying one row by 3 multiplies the determinant by 3.
How can the eigenvalues of a matrix A be found?
The eigenvalues are the solutions to the characteristic equation det(A−λI)=0
What is the difference between row-echelon form (REF) and reduced row-echelon form (RREF)?
REF has leading entries with zeros below them; RREF also has each leading 1 as the only nonzero entry in its column.
What is the determinant of an upper-triangular or lower-triangular matrix?
The determinant of a triangular matrix is equal to the product of its diagonal entries.
What is the geometric meaning of an eigenvector?
An eigenvector is a direction that is unchanged by the transformation — the vector may be stretched or flipped, but it does not change direction.
Explain how row operations relate to finding the inverse of a matrix, and describe what happens if a pivot cannot be found during the process.
To find the inverse of a matrix A, we perform row operations on the augmented matrix [A∣I] until the left side becomes the identity matrix. The right side then becomes A^{-1}. If at any step a pivot cannot be found (i.e., a zero column appears where a pivot is required), then A is singular and has no inverse.
If the determinant of a matrix A equals zero, what does this tell us about A?
It means that A is singular, which means it is not invertible and the corresponding linear transformation collapses space (it maps some nonzero vector to the zero vector).
If a matrix has distinct eigenvalues, what can be said about its eigenvectors?
The eigenvectors corresponding to distinct eigenvalues are linearly independent.
How are elementary matrices related to row operations?
Each elementary row operation can be represented by multiplying the matrix on the left by an elementary matrix. Thus, performing a row operation is equivalent to left-multiplying by an appropriate elementary matrix.
If two rows of a square matrix are identical, what can you conclude about its determinant, and why?
If two rows of a square matrix are identical, then the determinant of the matrix is zero. This is because the rows are linearly dependent, meaning the volume or area represented by the matrix collapses to zero — there’s no unique scaling or transformation possible.
If a 3×3 matrix A has an eigenvalue equal to zero, what does this imply about A?
It means that A is not invertible, since a zero eigenvalue implies that det(A)=0 and the transformation collapses at least one dimension.