Core Pure 1 Chapter 6: Matrices
26 flashcards · Mathematics · by TheoTyson
What is a matrix?
Array of elements.
What is a square matrix?
A matrix where the number of rows and columns is the same.
What is a zero matrix?
A matrix where all of the elements are zero.
What is an identity matrix (I)(I)(I) ?
Elements in the leading diagonal are 1, and the remaining elements are 0.
What is the notation for an identity matrix?
IkI_kIk where kkk is the size.
How do you add and subtract matrices? What is the condition that needs to be met?
Add or subtract the corresponding element in each matrix. Only add or subtract matrices that are the same size.
How do you multiply a matrix by a scalar?
Multiply every element in the matrix by the scalar.
What condition needs to be met for matrix multiplication?
The number of columns of the first matrix is equal to the number of rows in the second matrix.
Product Matrix Size
Same number of rows as the first matrix, and the same number of columns as the second matrix.
How do you multiply two matrices?
Multiply the elements in each row in the left hand matrix by the elements in each column in the right hand matrix.
What is the determinant of a 2 x 2 matrix M=abcdM = \begin{matrix} a & b \\ c & d \end{matrix}M=acbd ?
Det{\operatorname {Det}}Det MMM =ad−bc= ad - bc=ad−bc
What is the determinant of a 3 x 3 matrix M=M =M= abcdefghi\begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix}adgbehcfi ?
Det{\operatorname {Det}}Det MMM === a(ei−fh)−b(di−fg)+c(dh−eg)a(ei-fh) - b(di-fg) + c(dh-eg)a(ei−fh)−b(di−fg)+c(dh−eg)
What is the determinant of a singular matrix?
Det{\operatorname {Det}} Det MMM === 000
What is the determinant of a non-singular matrix?
Det{\operatorname {Det}}Det MMM  /=\mathrlap{\,/}{=}/= 000
What is the inverse of a 2 x 2 matrix MMM === abcd\begin{matrix} a & b \\ c & d \end{matrix}acbd ?
M−1M^{-1}M−1 === 1Det M\frac{1}{ {\operatorname {Det}} \, M}DetM1 ∗*∗ d−b−ca\begin{matrix} d & -b \\ -c & a \end{matrix}d−c−ba
Expand (AB)−1(AB)^{-1}(AB)−1
B−1B^{-1}B−1 A−1A^{-1}A−1
What is the transpose of a matrix?
MTM^TMT is found by interchanging the rows and columns.
How do you inverse a 3 x 3 matrix MMM === abcdefghi\begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix}adgbehcfi ?
Find the determinant of AAA , Det{\operatorname {Det}}Det AAA . Form a matrix of minors of AAA , MMM . Form the matrix of cofactors by changing the signs, CCC . +−+−+−+−+\begin{matrix} + & - & + \\ - & + & - \\ + & - & + \end{matrix}+−+−+−+−+ Form the transpose of the matrix, CTC^TCT . The inverse of matrix AAA is given by the formula A−1A^{-1}A−1 === 1Det A\frac{1}{ {\operatorname {Det}} \, A}DetA1 CTC^TCT
If AAA ⋅\cdotp⋅ xyx\begin{matrix} x \\ y \\ x \end{matrix}xyx === vvv , then what is xyx\begin{matrix} x \\ y \\ x \end{matrix}xyx equal to?
xyz\begin{matrix} x \\ y \\ z \end{matrix}xyz === A−1A^{-1}A−1 ⋅\cdotp⋅ vvv
What are consistent equations?
At least one set of values that satisfies all of the equations simultaneously.
What are inconsistent equations?
There are no sets of values that satisfy all of the equations simultaneously.
Describe the geometric interpretation when the equations form a non-singular matrix
The planes meet at a point. The system of equations is consistent. There is only one solution.
Describe the geometric interpretation when the equations are the same
All three equations represent the same plane. The system of equations is consistent. There are infinitely many solutions.
Describe the geometric interpretation when the equations are linear multiples
Two or more of the planes are parallel and non-identical. The system of equations is inconsistent. There are no solutions.
Describe the geometric interpretation when the equations are consistent
The planes form a sheaf. The system of equation is consistent. There are infinitely many solutions.
Describe the geometric interpretation when the equations are not consistent
The planes form a prism. The system of equations is inconsistent. There are no solutions.