MA265 Purdue Linear Algebra Flashcards

Effect of Elementary Row Operations on Determinant

• Multiplying a row by a scalar ( a ) multiplies the determinant by ( a ).
• Adding a multiple of one row to another row does not affect the determinant.
• Swapping two rows changes the sign of the determinant.

Inverse of 3x3 Matrix

• Calculate the determinant of matrix ( A ).
• Compute minors of ( A ) for each element.
• Apply sign pattern:
• Transpose the matrix of cofactors to obtain the adjugate.
• Divide the adjugate by the determinant to get the inverse.

Find the Adjugate (3x3)

• Construct the matrix of minors for matrix ( A ).
• Apply the alternating sign pattern to generate the matrix of cofactors.
• Transpose the matrix of cofactors to find the adjugate.

Linear Transformation

• Must maintain the form ( Ax ) and cannot include exponents or constants.

Nullity Fallacies

• Nullity equals the number of nonzero rows in row echelon form.
• Nullity does not equal the dimension of the column space.
• Nullity is not the same as the nullity of the transposed matrix.

Square Matrix Fallacies

• If matrix ( B ) is obtained from ( A ) through row operations, both share the same eigenvalue ( \lambda ).
• Determinant of the sum ( \text{det}(A+B) ) does not equal ( \text{det}(A) + \text{det}(B) ).
• If ( A ) and ( B ) are both nonsingular, their sum ( A+B ) may not be nonsingular.

Real Square Matrix Truths

• If ( \lambda ) is an eigenvalue of ( A ), then ( -\lambda ) is an eigenvalue of ( -A ).
• Any eigenvector ( v ) for ( A ) is also an eigenvector for ( 2A ).

Linear Independence

• A set of vectors is linearly independent if each column in reduced row echelon form (Rref) contains a leading one as its only nonzero entry.

The Following Are the Same

• The nullspace of matrix ( A ) is identical to the solution space of ( AX=0 ) and the orthogonal complement of the row space of ( A ).

Rank

• Defined as the number of leading 1's present in the Rref of matrix ( A ).

Nullity

• The number of columns in ( A ) that do not have leading 1’s in Rref represents the nullity.

To Test Whether a Vector ( v ) is in the Span of Vectors ( w_1, \ldots, w_k )

• Formulate the matrix ( A ) with ( w_1, \ldots, w_k ) as columns and check if ( AX=v ) maintains consistency.

To Test Whether Vectors ( w_1, \ldots, w_k ) Span the Vector Space ( V )

• Create matrix ( A ) with ( w_1, \ldots, w_k ) as columns and perform Rref. If every row contains a leading 1, then they span ( V ).

Square ( n \times n ) Matrix ( A )

• An invertible square matrix has a non-zero determinant, rank ( n ), linearly independent rows and columns, yielding exactly one solution for all ( b ), and a nullity of zero.

Determinant of ( kA )

• The determinant of matrix ( A ) multiplied by ( k ) is ( \text{det}(A) \times k^n ), where ( n ) is the number of rows in ( A ).

Area of a Triangle with Corners ((x_1,y_1), (x_2,y_2), (x_3,y_3))

• The area can be calculated using the determinant of the matrix: [ \begin{bmatrix} 1 & 1 & 1 \ x_1 & x_2 & x_3 \ y_1 & y_2 & y_3 \end{bmatrix} ]