Matrix Size
The element:
a_12
In the matrix (assuming it's not zero-indexed):
[[1,2,3],[4,5,6],[7,8,9]]
What is 2?
The transpose of:
[[1,2],[3,4]]
What is:
[[1,3],[2,4]]
(Upper/Lower/Neither/Both) The following is a type of matrix from the triangular family of matrices:
[[1,0,0],[2,1,0],[3,1,0]]
What is lower triangular?
A+B, where:
A = [[1,2]], B = [[3,4]]
What is:
[[4,6]]
?
(T/F) With reasoning:
tr(diag(d_1,d_2,...,d_n)) = d_1 + d_2 + ... + d_n
What is True?
(Reasoning: the diagonal matrix has diagonal entries d_1 .... d_n, so the trace is the sum of those entries)
The size of the matrix:
[[-1,7],[2,5],[-1,0]]
What is 3x2?
The trace of:
[[-1,0,2],[0,1,2],[1,2,7]]
What is 7?
(Upper/Lower/Neither/Both) The following is a type of matrix from the triangular family of matrices:
[[0,1,1],[1,0,1], [0,0,1]]
What is neither?
3A, where
A = [[1,3,5], [-2,5,-7],[2,0,1]]
What is:
[[3, 9, 15],[-6,15,-21],[6,0,3]]
?
(T/F) To be symmetric, we just require:
A = A^T
What is true?
The 3rd row vector from the matrix:
[[1,2],[-1,-2],[-4,5],[2,4]]
What is:
\vec{a_3} = [[-4, 5]]
?
The transpose of:
[[a,b,c],[d,e,f],[g,h,i],[j,k,l]]
What is:
[[a,d,g,j],[b,e,h,k],[c,f,i,l]]
?
(Upper/Lower/Neither/Both) The following is a type of matrix from the triangular family of matrices:
[[1,0,0],[0,-5,0],[0,0,3]]
What is both upper/lower triangular?
3A - 2B, where:
A = [[1,2],[-1,-2]], B = [[1,2],[3,4]]
What is:
[[1, 2], [-9,-14]]
?
(T/F) The following matrix is diagonal:
[[0,0,0], [0,0,0],[0,0,0]]
What is true?
(Diagonal just requires upper and lower triangular)
The 4-th column vector from the matrix:
[[1,2,4,5,6,7]]
What is:
\vec{b_4} = [5]
?
The trace of:
[[a,b,c],[d,e,f],[g,h,i],[j,k,l]]
What is undefined?
(trace is only defined for square matrices)
The matrix function's domain:
A(t) = [[t,t^{1/2}],[\sin(t),cos(t)]]
What is all positive reals (including 0)?
3A - 2B where:
A = [[1,2],[-1,-2]], B = [[1,2],[3,4], [5,6]]
What is undefined?
(The addition of a 2x2 and 3x2 matrix is undefined)
What is true?
(True/False) Two matrices equal if they are both mxn in size.
(We also require their entries to be the same)
(True/False) The trace of an mxn matrix A is always defined, and can be calculated as:
\tr(A) = a_11 + a_22 + ... + a_{ii}
What is False?
(The matrix MUST be a square matrix for this to be correct)
(True/False) The square (nxn) matrix that is both upper and lower triangular, and has:
tr(M) = 0
Is the nxn zero matrix
What is False?
(The diagonal could alternate (ex: 1 - 1 + 1 - 1 + ...) and thus have non-zeroes on its diagonal)
(True/False) The scalar multiplication of an mxn matrix is always defined.
What is true?
(Only addition/subtraction and later matrix mult. have dimension requirements)
(T/F) Suppose that A is an nxn matrix who is both upper and lower triangular. A then is already symmetric, and is skew-symmetric iff tr(A) = 0.
What is false?
(A would be already symmetric, but we again could have a trace such that diag(1,-1,1,-1,....,1) on the diagonal of A).