Let A and C be a pair of matrix where the product AC exists. 1. Show...

Let A and C be a pair of matrix where the product AC exists.

1. Show that rank(AC) ≤ rank(A)

2. Give an example such that rank(AC) < rank(A)

3. is rank(AC) ≤ rank(C) always true? if not give a counterexample.

Expert Solution

Colby Messinger answered 2 years ago

Matrix A belongs to an n×n matrix over F. show that there exists a nonzero polynomial...

Matrix A belongs to an n×n matrix over F. show that there exists a nonzero polynomial f(x) belongs to F[x] such that f(A) ＝0.

Show that R-1(a)R(a) = I, where I is the identity matrix and R(a) is the rotation matrix.

Show that R-1(a)R(a) = I, where I is the identity matrix and R(a) is the rotation matrix. This equation shows that the inverse coordinate transformation returns you to the original coordinate system.

Prove: Let A be an mxm nonnegative definite matrix with rank(A)=r Then there exists an mxr...

Prove: Let A be an mxm nonnegative definite matrix with rank(A)=r Then there exists an mxr matrix B having rank of r, such that A=BBT

Let C be the following matrix: C=( 1 2 3 -2 0 1 1 -2 -1...

Let C be the following matrix: C=( 1 2 3 -2 0 1 1 -2 -1 3 2 -8 -1 -2 -3 2 ) Give a basis for the row space of Cin the format [1,2,3],[3,4,5], for example.

Given a plant with the genotype C/cm; Ac+/Ac+, where cm is an unstable allele caused by...

Given a plant with the genotype C/cm; Ac+/Ac+, where cm is an unstable allele caused by a Ds insertion. This plant is crossed with another plant with the genotype C/cm where cm is an unstable allele caused by Ac insertion. What phenotypes would be produced (including their proportion)? Assume that Ac and c are unlinked, that the chromosome-breakage frequency is negligible, and that mutant c/C is Ac+.

Suppose that ac + bd = 0. Show that vectors [a, b] and [c, d] are...

Suppose that ac + bd = 0. Show that vectors [a, b] and [c, d] are perpendicular. The number ac + bd is called the dot product of the vectors [a, b] and [c, d].

Let A be a 2×2 symmetric matrix. Show that if det A > 0 and trace(A)...

Let A be a 2×2 symmetric matrix. Show that if det A > 0 and trace(A) > 0 then A is positive definite. (trace of a matrix is sum of all diagonal entires.)

Prove Desargues’ Theorem for the case where AC is parallel with A′C′ on the extended Euclidean...

Prove Desargues’ Theorem for the case where AC is parallel with A′C′ on the extended Euclidean plane.

Consider the following AC circuit. Let R = 1.0kΩ, C = 150µF and L = 10.0...

Consider the following AC circuit. Let R = 1.0kΩ, C = 150µF and L = 10.0 mH. The AC voltage source is of frequency 60 HZ and has a maximum voltage Vmax of 100 V. Find a) the capacitive reactance, b) inductive reactance, c) total impedance, d) RMS current, e) voltage drop across the inductor, f) voltage drop across the capacitor, g) phase angle, h) draw the phasor diagram for all voltage drops and the resulting phasor, and i) is...

real Analysis 1). Let C be the intersection of all the Cn's. You must show C...

real Analysis 1). Let C be the intersection of all the Cn's. You must show C is closed, measurable, has positive measure, and contains no interval, i.e., if x is in C, every epsilon neighborhood of x contains points not in C.Analysis i should have note 0<alpha<1 Start with the interval [0,1] and remove the middle open interval of length alpha/3 to form C1. Cn is then formed by removing the middle open interval of length alpha/3^n from each closed...

Question