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# 数学代考|线性代数代写Linear algebra代考|MATH7000 Column Operations

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## 数学代考|线性代数代写LINEAR ALGEBRA代考|Column Operations

We can perform operations on the columns of a matrix in a way that is analogous to the row operations we have considered. The next theorem shows that column operations have the same effects on determinants as row operations.

Remark: The Principle of Mathematical Induction says the following: Let $P(n)$ be a statement that is either true or false for each natural number $n$. Then $P(n)$ is true for all $n \geq 1$ provided that $P(1)$ is true, and for each natural number $k$, if $P(k)$ is true, then $P(k+1)$ is true. The Principle of Mathematical Induction is used to prove the next theorem.
If $A$ is an $n \times n$ matrix, then $\operatorname{det} A^{T}=\operatorname{det} A$.
PROOF The theorem is obvious for $n=1$. Suppose the theorem is true for $k \times k$ determinants and let $n=k+1$. Then the cofactor of $a_{1 j}$ in $A$ equals the cofactor of $a_{j 1}$ in $A^{T}$, because the cofactors involve $k \times k$ determinants. Hence the cofactor expansion of $\operatorname{det} A$ along the first row equals the cofactor expansion of $\operatorname{det} A^{T}$ down the first column. That is, $A$ and $A^{T}$ have equal determinants. The theorem is true for $n=1$, and the truth of the theorem for one value of $n$ implies its truth for the next value of $n$. By the Principle of Mathematical Induction, the theorem is true for all $n \geq 1$.

Because of Theorem 5, each statement in Theorem 3 is true when the word row is replaced everywhere by column. To verify this property, one merely applies the original Theorem 3 to $A^{T}$. A row operation on $A^{T}$ amounts to a column operation on $A$.

Column operations are useful for both theoretical purposes and hand computations. However, for simplicity we’ll perform only row operations in numerical calculations.

## 数学代考|线性代数代写LINEAR ALGEBRA代考|Determinants and Matrix Products

The proof of the following useful theorem is at the end of the section. Applications are in the exercises.
Multiplicative Property
If $A$ and $B$ are $n \times n$ matrices, then $\operatorname{det} A B=(\operatorname{det} A)(\operatorname{det} B)$.
EXAMPLE 5 Verify Theorem 6 for $A=\left[\begin{array}{ll}6 & 1 \ 3 & 2\end{array}\right]$ and $B=\left[\begin{array}{ll}4 & 3 \ 1 & 2\end{array}\right]$ SOLUTION
$$A B=\left[\begin{array}{ll} 6 & 1 \ 3 & 2 \end{array}\right]\left[\begin{array}{ll} 4 & 3 \ 1 & 2 \end{array}\right]=\left[\begin{array}{ll} 25 & 20 \ 14 & 13 \end{array}\right]$$
and
$$\operatorname{det} A B=25 \cdot 13-20 \cdot 14=325-280=45$$
Since $\operatorname{det} A=9$ and $\operatorname{det} B=5$,
$$(\operatorname{det} A)(\operatorname{det} B)=9 \cdot 5=45=\operatorname{det} A B$$
Warning: A common misconception is that Theorem 6 has an analogue for sums of matrices. However, $\operatorname{det}(A+B)$ is not equal to $\operatorname{det} A+\operatorname{det} B$, in general.

## 数学代考|线性代数代写LINEAR ALGEBRA代考|Determinants and Matrix Products

If一个和乙是n×n矩阵，然后这一个乙=(这一个)(这乙).

(这一个)(这乙)=9⋅5=45=这一个乙

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