An industrial system with three industries has the input-output matrix D and external demand matrix E below. Solve for the output matrix X in the equation X = DX + E. D=begin{bmatrix}0.1 & 0.3&0.2 0.0 & 0.2&0.30.4&0.1&0.1 end{bmatrix} text{ and } E=begin{bmatrix}3000 35008500 end{bmatrix}

Find a basis for the space of $2\times 2$ diagonal matrices.
$\text{Basis }=\{\left[\begin{array}{cc}& \\ & \end{array}\right],\left[\begin{array}{cc}& \\ & \end{array}\right]\}$

Let B be a 4x4 matrix to which we apply the following operations:
1. double column 1,
2. halve row 3,
3. add row 3 to row 1,
4. interchange columns 1 and 4,
5. subtract row 2 from each of the other rows,
6. replace column 4 by column 3,
7. delete column 1 (column dimension is reduced by 1).
(a) Write the result as a product of eight matrices.
(b) Write it again as a product of ABC (same B) of three matrices.

Find the characteristic polynomial of the matrices
$\left[\begin{array}{cc}2& 1\\ -1& 3\end{array}\right]$

Perform the indicated matrix operations B - A given that A, B, and C are defined as follows. If an operation is not defined, state the reason.
$A=\left[\begin{array}{cc}4& 0\\ -3& 5\\ 0& 1\end{array}\right]B=\left[\begin{array}{cc}5& 1\\ -2& -2\end{array}\right]C=\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]$

Describe how to subtract matrices.

Penguins spend almost 75% of their lives in the sea. An Emperor Penguin in the wild has a life span of about 18 years. About how many years does this penguin spend In the sea?

How these pair are taken?What is the rule behind it?