Linear Algebra - Chapter 2 "Matrix Operations"
ok
!Must be able to write transpose on test!
linear combination (linearly dependent)
"In the span" means...
X, Y, and Z should be isolated on the left side of the equal sign, and A, B, and C should be on the right side of the equal sign, not necessarily isolated. ( Example: X = I Y = I - BC Z = 0 )
"Solve for X, Y, and Z in terms of A, B, and C." In the answer, ___, ___, and ___ should be isolated on the left side of the equal sign, and ___, ___, and ___ should be on the right side of the equal sign, not necessarily isolated---may have Identity and null matrices with them.
not a linear combination (linearly independent)
"spans all real numbers" means...
left singular (doesn't have an inverse)
(Finding inverse of a matrix) If the ___ side ends up with a row of all zeros during reducing, then the matrix A is ___
True
(T / F) Any square matrix that has a column of all zeros is singular.
false (tricky one. x does = A⁻¹b, but not bA⁻¹. Remember that order is very important in linear algebra. Think about it this way: b is by definition a vertical vector, so its dimensions are #x1. If A is an invertible matrix then it must have more than one row, which would make the product of bA⁻¹ undefined: bA⁻¹ = #x1 * (>1)x(>1) = undefined (not possible) however... A⁻¹b = mxn * nx1 is possible. )
(T / F) Ax = b: if A is invertible, then x = bA⁻¹
True
(T or F) (5I₃)A is the same as 5A (when A is any matrix with 3 rows yes, even a 3x200 matrix would work with this)
True
(T or F) A*Iₙ = Iₙ*A = A
False ( A zero matrix is just a matrix with any dimensions that has all elements inside the matrix as 0. It does NOT have to be a square matrix. Non-square zero matrix: [0 0] [0 0] [0 0] )
(T or F) All zero matrices, like diagonal and identity matrices, must also be square matrices.
True
(T or F) You can make an augmented matrix with A as an invertible coefficient matrix and any value of ^b, and you will get a distinct (unique) solution.
False (Diagonal matrices can have non-1's in the main diagonal, making them row echelon only)
(T or F) All diagonal matrices are in reduced row echelon form.
True
(T or F) All identity matrices are in reduced row echelon form.
True (mult each row by something, it doesn't affect the other rows because of the one pivit point and all others are 0)
(T or F) An I matrix can be used to create the components of any vector.
False (square matrices can be added, multiplied, and subtracted)
(T or F) Square matrices with the same number of rows and columns can be added and multiplied together, but not subtracted.
False (square matrices can be added and multiplied)
(T or F) Square matrices with the same number of rows and columns can be added together, but not multiplied.
False (I'm not even gonna explain this one, come on.)
(T/F) If A is not invertible, there is a matrix D such that AD = I
False (They could be invertible, but just not the inverses of each other necessarily.)
(T/F) If AB ≠ I, then both A and B must be singular.
True
(T/F) The coefficient matrix of a linear system is invertible, so the system is consistent.
True*******not sure yet, could also have infinite solutions*******
(T/F) The coefficient matrix of a linear system is singular, so the system is inconsistent.
False (Non-square matrices are incapable of having an inverse---true, but the term "singular" only applies to square matrices that do not have an inverse)
(T/F) All non-square matrices are singular.
False (because if they are linearly independent, then they cannot be used to create each other, which means they cannot create, span, all real numbers *not super sure bout this one, but I think that's why*)
(T/F) If the columns of A are linearly independent, then the columns of A span R. (Where R is all real numbers)
singular
*Extra: if a square matrix has a determinate of 0, then it must be...
invertible
*Extra: if a square matrix has a determinate that is nonzero, then it must be...
Inverse of a square matrix
- Not all (square) matrices have an inverse - When a square matrix doesn't have an inverse, it is said to be "singular" Inverse matrices: AB = BA + I A⁻¹ = B AA⁻¹ = I A⁻¹A = I
5
A 3x5 matrix has how many columns?
3
A 3x5 matrix has how many rows?
5
A 8x5 matrix has how many columns?
8
A 8x5 matrix has how many rows?
inverses.
A and A^-1 are ___ of each other?
transpose
A diagonal matrix is symmetric, that is, equal to its ___:
square
A diagonal matrix is what kind of matrix?
diagonal
A square zero matrix technically qualifies as a ___ matrix.
first (left) second (right)
A=[aij] is an m×n matrix and B=[bij] is an n×p matrix, the product AB is an m×p matrix. In other words, when you multiply two matrices, the resulting matrix will have the same number of rows as the ___ matrix and the same number of columns as the ___ matrix.
I
AA^-1 = I A^-1A = ???
They are inverses of each other
AB = BA = I coefficient matrix A times coefficient matrix B is equal to B times A (matrices having the communitive property is rare) both of these operations results in the identity matrix. What does that tell us about A and B?
size partitioned (in other words, just like regular matrices, you can't add two partitioned matrices that do not have identical dimensions)
Addition and Scalar multiplication of partition matrices must involve matrices that are of the same ___ and ___ in the same way. (parts must be the same size)
row echelon reduced row echelon (if it is also an identity matrix)
All diagonal matrices are in ___ form, and may also be in ___ form.
partitioned matrix
Also called "block matrices," have components (entries) that are matrices. (Matrices inside a matrix---it's like Inception meets the Matrix up in here)
diagonal square
An identity matrix is a type of ___ matrix, which in turn means it must also be a ___ matrix.
matrix
An mxn ___ is a rectangular array of numbers arranged in m rows and n columns.
square
Any two ___ matrices of the same order can be added and multiplied.
distinct A^(-1)*b (The inverse of A times b)
Ax = b: if A is invertible, then the system has a ___ solution, and x = ?
False (The diagonal matrix is not always an identity matrix. Second question is irrelevant and inaccurate.)
Determine the validity of the following statement. If it is true, answer the second question. 1.) The diagonal matrix is always an identity matrix, but not all identity matrices are diagonal matrices. 2.) The reason not all identity matrices are diagonal matrices is...?
(1.) True (2.) A diagonal matrix can have non-ones along its main diagonal, whereas the identity matrix is characterized by having only 1's along its main diagonal. ( Diagonal, but not Identity: [5 0 0] [0 2 0] [0 0 -1] Diagonal and Identity: [1 0 0] [0 1 0] [0 0 1] )
Determine the validity of the following statement. If it is true, answer the second question. 1.) The identity matrix is always a diagonal matrix, but not all diagonal matrices are identity matrices. 2.) The reason not all diagonal matrices are identity matrices is...?
matrix factorization
Expresses a given matrix as a product of 2 or more others. Frequently used when the given matrix is involved in numerous solutions. (Take one matrix, break it into more matrices)
square
For a given matrix to have an inverse, it must be...
one
Functionally, the identity matrix is a bit like a ___
2x5
Give the dimensions of the resulting matrix. 2x3 * 3x5
436x537
Give the dimensions of the resulting matrix. 436x335 * 335x537
undefined
Give the dimensions of the resulting matrix. 6x2 * 3x6
7x2
Give the dimensions of the resulting matrix. 7x5 * 5x2
Yes
I*A = A*I but is: 4I*A = A*4I ?
True / ok
INvertible means INdependent
L(U^x) = ^b
If A = LU, then A^x = ^b can be written as ___ which makes for easier solution since L and U are triangular matrices.
1. Has only the trivial solution (The inverse of a matrix is distinct, which means each variable has only one value that it equals. With a homogeneous augmented matrix the only possibilities are trivial (which is distinct - each variable = 0) or infinite solutions if a free variable is present. Clearly, this means if the the matrix A is invertible, it would have the trivial solution when set = to a zero vector ^0)
If A is invertible, A^x = ^0 1. Has only the trivial solution 2. Has more than the trivial solution (infinite solutions)
2 (one solution)
If A is invertible, A^x = ^b has ___ for any ^b 1. Infinite solutions 2. One solution 3. No solution
1. row equivalent (which if you'll remember means there is a set of elementary row operations that will turn A into I)
If A is invertible, it is ___ to the identity matrix 1. row equivalent 2. not row equivalent
independent (This is a bit like the other definition with the ^0 vector. Remember, independence is trivial, so what this is saying is if you put a column of zeros on the right hand side of a invertible matrix and solve, you'll get the trivial solution, which means it's independent)
If A is invertible, the columns of A form a linearly ___ set.
also invertible (the inverse matrix of A's transpose is the transpose of A's inverse.)
If A is invertible, then A^T is ___
inverse of A
If A is invertible, there is a matrix C such that CA = I In this case, C is the ___
inverse of A
If A is invertible, there is a matrix D such that AD = I In this case, D is the ___
2. Has more than the trivial solution (infinite solutions)
If A is singular, A^x = ^0 1. Has only the trivial solution 2. Has more than the trivial solution (infinite solutions)
1 (infinite solutions) or 3 (No solution)
If A is singular, A^x = ^b has ___ for any ^b 1. Infinite solutions 2. One solution 3. No solution
No (Because then C would have to be the inverse of A, and since A is singular, we know there is no inverse.)
If A is singular, is there a matrix C such that CA = I?
2. not row equivalent (elementary row operations cannot turn A into I if A is singular)
If A is singular, it is ___ to the identity matrix 1. row equivalent 2. not row equivalent
dependent (Remember, dependence means you have (a) free variable(s), so what this is saying is if you put a column of zeros on the right hand side of a singular matrix and solve, you'll get a row of all zeros, which means it's linearly dependent)
If A is singular, the columns of A form a linearly ___ set.
also singular
If A is singular, then A^T is ___
invertible (they're inverses of each other, that's why you get the I matrix when you multiply them)
If AB = I, then both A and B are ___
equal
If two matrices are the exact same dimensions and have the exact same corresponding components, (in other words, if they are mirror images of each other) they are...?
infinite or inconsistent (no solution)
If you reduce a singular matrix, you will get what kind of solution(s)?
distinct
If you reduce an invertible matrix, you will get what kind of solution(s)?
Create an augmented matrix with all the equivalent vectors on the right of the coefficient matrix and solve (RRE). The variables will line up with their values for each equivalent vector.
If you're given a coefficient matrix and several equivalent vectors and asked to find the values of the variables for each vector, what should you do?
the matrices must have the same dimensions ( A = 2x3 and B = 2x3 A = 3x3 and B = 3x3 A = 100x100 and B = 100x100 etc. )
In order for two matrices to be added or subtracted...?
columns rows
In order for two matrices to be multiplied, the number of ___ in the first matrix must be equal to the number of ___ in the second matrix.
singular (does not have an inverse)
In terms of invertible matrices, Span, linear combination, and dependent all mean the matrix is ___
invertible
In terms of invertible matrices, not in span, not a linear combination, and independent all mean the matrix is ___
Transpose of a Matrix
Interchange the rows and columns; aᵢⱼ becomes aⱼᵢ
Yes, both.
Is a diagonal matrix an upper or lower triangular matrix?
2 2
I₂ Would be an identity matrix with ___ rows and ___ columns
identity matrix
Iₙ - a diagonal matrix with all 1's on the diagonal; when multiplied to a matrix, regardless of order, nothing changes.
capital letters (A, B, C etc.)
Matrices are represented by...
order is very specific
Matrix products are not commutative (AB does not = BA) therefore, ___.
augmented matrix with A on left and I on right reduced row echelon form The right side will be the inverse (if the left side ends up with a row of all zeros during reducing, there is no inverse and the matrix A is singular.)
Method for finding inverse:
ok
Method for finding inverse: 1.) Set up an augmented matrix such that the left half is A and the right half is an identity matrix with the same number of rows and columns as A. 2.) Reduce the augmented matrix into reduced row echelon form. 3.) Once in RRE form, the right side of the matrix will be the inverse (A⁻¹), and the left side will be the identity. (They kinda swap places.) Note: If the left side of the augmented matrix has a row of all zeros, then the matrix A is singular (doesn't have an inverse).
equal (Make them match)
Multiplication of partitioned matrices has the same rule as for regular matrices, that is column partitions of A must be ___ to row partitions of B.
inverse
Not all (square) matrices have an ___
2 [ 4x2 4x2 ]
Observe the following matrices and determine which ones could be added or subtracted successfully to a 4x2 matrix. 1. 4x4 2. 4x2 3. 2x4 4. 1x5 5. 3x7
2 and 5 [ 2x3 * 3x3 2x3 * 3x7 ]
Observe the following matrices and determine which ones could be multiplied successfully to a 2x3 matrix, A, if A is on the left side. 1. 4x3 2. 3x3 3. 2x3 4. 1x5 5. 3x7
1 and 3 [ 4x4 * 4x2 2x4 * 4x2 ]
Observe the following matrices and determine which ones could be multiplied successfully to a 4x2 matrix, A, if A is on the right side. 1. 4x4 2. 4x2 3. 2x4 4. 1x5 5. 3x7
ok
Professional software for high-performance numerical linear algebra, such as LAPACK, makes intensive use of partitioned matrix calculations.
No (Surprisingly, multiplying by a "one matrix" will give you different results depending on which side it's on.)
Say A is a 3x3 matrix consisting of all ones, and B is some non-Identity 3x3 matrix. (See image for example) The result of dot producting AB is C. Question, will BA = this C as well? (Basically, does multiplying by a matrix of all ones that is the same size obey commutative property?)
No, because all inverses are distinct. (There can only be one inverse per matrix. You can't have AB = BA = I AND AC = CA = I unless B and C are the exact same matrix.)
Say we have three invertible matrices that are all the same size. Call them A, B, and C. Now, let's say: B ≠ C AB = BA = I Is it possible for C to be the inverse of A? Justify your answer.
Yes
See image. Are the two matrices equal?
Yes ( At first glance they appear different, but if you calculate the actual values of the components, you'll find that they are equivalent. 1/2 = 0.5 sqrt(16) = 4 )
See image. Are the two matrices equal?
3 -1 6
See image. Give the components that correspond to the following addresses in matrix A: A: a₁₂ A: a₃₁ A: a₂₂
0 2 -2
See image. Give the components that correspond to the following addresses in matrix B: B: a₂₁ B: a₃₂ B: a₁₃
4 0 8
See image. Give the components that correspond to the following addresses in matrix C: C: a₁₁ C: a₃₃ C: a₂₃
2 8 7
See image. Give the components that correspond to the following addresses in matrix D: D: a₃₁ D: a₂₃ D: a₁₃
singular (tldr: having two identical columns means it will reduce into a row of all zeros and not have a distinct solution, making it singular. Full explanation: Two of the columns are identical, therefore, they are linear combinations of each other: 1*1st column = 2nd column if they are linear combinations, that means they are also dependent. If you remember, the dependent solution has a row of all zeros on the bottom: 0 0 0 Remember we're trying to determine whether this matrix has an inverse or not. Now, invertible matrices row reduce into the identity matrix, which would have a bottom row that looks like this if it's 3x3: 0 0 1. As you can see, simply knowing that the 1st and 2nd columns are linear combinations of each other tells us a great deal about the properties of the matrix. In this case, it tells us that when reduced, the matrix will have infinite solutions, be dependent, and have more than the trivial solution, and most importantly, will not reduce into the identity matrix. This of course means it is not invertible.)
See image. Is the matrix invertible or singular? (Don't reduce, you should be able to tell just by looking at it)
Yes (There's no rule about what the values of the main diagonal must be, only that the non-diagonal values must be 0)
See the image. Does that qualify as a diagonal matrix?
vector pipeline
Some high-speed computers, particularly those with ___ architecture, perform matrix calculations more efficiently when the algorithms use partitioned matrices.
partitioned matrix
The following is an example of which type of matrix? [A B] [C D]
distinct (unique) (This means that there is only one distinct ordering of numbers that can be the inverse of a matrix)
The inverse of a matrix is ___
Addition of Matrices
The matrices must be the same size, and then corresponding components are added.
True
The product of A (when A is a 2x2 matrix) and an I₂ identity matrix will equal A.
False ( The multiplication would not be possible because the identity matrix I₃ is: [1 0 0] [0 1 0] [0 0 1] And matrix A is: [# #] [# #] The columns of I₃ are not = to the rows of A, so the product is undefined. )
The product of A (when A is a 2x2 matrix) and an I₃ identity matrix will be equivalent to A.
ok
To check whether two matrices can be multiplied, compare the columns to the rows. When actually performing the multiplication, multiply the rows by the columns.
multiply an identity matrix
To determine whether two matrices are inverses of each other, ___ them. If they are, the result will be ___.
A coefficent matrix times its inverse results in the Identity matrix
Translate the following formula into words: AA^-1 = I
NEVER. EVER. NO SUCH THING.
Under what circumstances can two matrices be divided?
Partitioned Matrices
Used as abbreviations for very large matrices.
singular (or degenerate)
When a matrix doesn't have an inverse, it is said to be ___
nothing changes
When an identity matrix is multiplied to a matrix, regardless of order, ___
inverse
When looking at inverse matrix problems, make sure you're not just being asked to find whether a matrix has an ___ or not. If that's all it's asking, you just need the matrix in row echelon form, not row reduced ech. form.
partitioning (For instance, one linear programming research team simplifies a problem by partitioning the matrix into 837 rows and 51 columns. The problem's solution took about 4 minutes on a Cray supercomputer.)
When matrices are too large to fit in a computer's high-speed memory, ___ permits the computer to work with only two or three sub-matrices at a time.
diagonal
When transposing a matrix, the ___ stays the same.
The left side.
When trying to determine whether a matrix is invertible which side should you be paying attention to?
itself
Where I is an identity matrix, I = Iᵀ. In other words, the transpose of an identity matrix is equal to ___.
LU Factorization
a given matrix's factorization into a lower triangular matrix and an upper triangular matrix.
Zero matrix
a matrix in which every element is zero
square matrix
a matrix with the same number of rows and columns
diagonal matrix
a square matrix with all nondiagonal components being zero.
ith row jth column
aᵢⱼ is the component that appears in the ___ and ___.
scalar multiplication of a matrix
multiplying any matrix by a constant called a scalar; the product of a scalar k and an m x n matrix - Every component is multiplied by the scalar
product of two matrices
the number of columns in the 1st matrix must match the number of rows in the 2nd matrix; rows of the 1st matrix dot product with columns of the 2nd matrix.
