To illustrate the multiplication of matrices, and also the fact that matrix multiplication is not necessarily commutative, consider the matrices A=begin{bmatrix}1 & -2&1 0 & 2&-12&1&1 end{bmatrix} B=begin{bmatrix}2 & 1&-1 1 & -1&02&-1&1 end{bmatrix}

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

simplify by combining like terms 4x-6x

Determine:
$\left[\begin{array}{ccc}1& -1& 2\end{array}\right]\{\left[\begin{array}{c}4\\ -1\\ 0\end{array}\right]\left[\begin{array}{cc}2& 3\end{array}\right]+\left[\begin{array}{cc}1& 0\\ 0& 1\\ -1& -1\end{array}\right]\}$

True or False - Determinants of two similar matrices are the same. Explain.

True or False If A and B are $n\times n$ lower triangular matrices, then AB is also lower triangular

 

Let B be a $4\times 4$ 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 (so that the column dimension is reduced by 1).
(a) Write the result as a product of eight matrices.
(b) Write it again as a product ABC (same B) of three matrices.