Math Problems Vault

Problem V.1.1

(a)(b)(c)

The matrix

\left\| \begin{array}{lll} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array}\right\| .

is given.

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(a)

Write out the submatrix located in rows 1 and 3 and columns 1 and 3.

(b)

How many square submatrices of second order does the given matrix have?

(c)

How many submatrices does it have in all?

Problem V.1.2

(a)(b)(c)(d)

The matrices

A=\left\| \begin{array}{lll} 1 & 2 & 3 \\ 4 & 5 & 6 \end{array}\right\| , \quad B=\left\| \begin{array}{ll} 2 & 3 \\ 5 & 6 \\ 8 & 9 \end{array}\right\| .

are given. Is it possible to add the matrices

?

(a)

$A$ and $B$

(b)

$A^{T}$ and $B$

(c)

$A$ and $B^{T}$

(d)

$A^{T}$ and $B^{T}$ ?

Problem V.1.3

The matrices

A=\left\| \begin{array}{ll} 1 & 1 \\ 1 & 2 \end{array}\right\| , \quad B=\left\| \begin{array}{cc} 2 & 1 \\ -1 & 1 \end{array}\right\| , \quad C=\left\| \begin{array}{ll} 4 & 3 \\ 1 & 5 \end{array}\right\| .

are given. Compute the matrix $2 A+3 B-C$ .

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Problem V.1.4

With what coefficients does the matrix

D=\left\| \begin{array}{ll} 1 & 2 \\ 4 & 5 \end{array}\right\|

expand in the matrices $A$ and $B$ and $C$ of the preceding problem?

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Problem V.1.5

(a)(b)

Is it possible to expand the matrix $\left\| \begin{array}{ll}1 & 3 \\ 7 & 9\end{array}\right\|$ in the matrices

?

(a)

$A$ and $B$ of problem 3

(b)

$A$ and $B$ and $C$ of problem 3?

Problem V.1.6

Are the rows

\mathbf{a}=\| 1 \quad 2 \quad 3 \quad 4\| , \quad \mathbf{b}=\| 2 \quad 3 \quad 4 \quad 5\| , \quad \mathbf{c}=\| 3 \quad 4 \quad 5 \quad 6\| ?

linearly independent?

?