Math Problems Vault

Problem 15.25

Prove that the $k$ -th column of the matrix $AB$ equals the product of the matrix $A$ and the $k$ -th column of $B$ .

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Problem 15.26

State and prove a proposition analogous to 15. 25, for rows.

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Problem 15.27

Prove that the $k$ -th column of the matrix $AB$ equals a linear combination of the columns of the matrix $A$ with coefficients from the elements of the $k$ -th column of the matrix $B$ .

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Problem 15.28

State and prove the analogue of proposition 15. 27 for rows.

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Problem 15.29

(1)(2)(3)

Prove that:

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(1)

when two rows of the matrix $A$ are interchanged, the corresponding rows in $AB$ are also interchanged

(2)

if the $k$ -th row of the matrix $A$ is multiplied by a number $\lambda$ , then the $k$ -th row of $AB$ is also multiplied by $\lambda$

(3)

if the $j$ -th row of the matrix $A$ is added to its $k$ -th row, then the same elementary transformation happens to the matrix $AB$ .

Problem 15.30

(1)(2)(3)

State and prove the analogues of propositions 15.29 for columns. Namely, prove that

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(1)

when two columns of the matrix $A$ are interchanged, the corresponding columns in $AB$ are also interchanged

(2)

if the $k$ -th column of the matrix $A$ is multiplied by a number $\lambda$ , then the $k$ -th column of $AB$ is also multiplied by $\lambda$

(3)

if the $j$ -th column of the matrix $A$ is added to its $k$ -th column, then the same elementary transformation happens to the matrix $AB$ .

Problem 15.31

(1)(2)

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(1)

Prove that adding to a row of a matrix a linear combination of its remaining rows can be accomplished by successively applying basic elementary row transformations.

(2)

Prove the analogous statement for the transformation consisting of interchanging two rows of a matrix.

Problem 15.32

Compute the product $\mathbf{e}_{i} A\mathbf{e}_{k}^{T}$ for an arbitrary matrix $A$ ( $\mathbf{e}_{i}$ denotes the $i$ -th row of the identity matrix of a suitable size).

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Problem 15.33

(1)(2)

For an arbitrary matrix $A$ and a matrix unit $E_{ij}$ of a suitable size, compute the product:

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(1)

$E_{ij} A$

(2)

$AE_{ij}$ .

Problem 15.34

Let the matrices $A$ and $B$ be such that for arbitrary columns $\xi$ and $\eta$ of a suitable height the equality $\xi^{T} A\eta = \xi^{T} B\eta$ holds. Prove that $A = B$ .

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Problem 15.35

Let $A$ be a matrix of dimensions $m \times n, E_{m}$ and $E_{n}$ identity matrices of order $m$ and $n$ respectively. Prove that $E_{m} A = AE_{n} = A$ .

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Problem 15.36

(1)(2)

By what matrix should the matrix $A$ be multiplied in order to obtain:

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(1)

the first column of $A$ ;

(2)

the first row of $A$ ?

Problem 15.37

(1)(2)(3)

Choose an elementary matrix $K$ such that the matrix $KA$ is obtained from $A$ :

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(1)

by interchanging the first two rows of $A$

(2)

by adding the first row to the second

(3)

by multiplying the first row of $A$ by a number $\lambda \neq 0$ .

Problem 15.38

Choose an elementary matrix $K$ such that the product $AK$ is obtained from $A$ by means of a given elementary transformation of columns.

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