Question

Given some numbers $\mathit{a}\mathbf{,}\mathit{b}\mathbf{,}\mathit{c}\mathbf{,}\mathit{d}$, $\mathit{e}$ and $\mathit{f}$ such that $\mathit{d}\mathit{e}\mathit{t}\left[\begin{array}{ccc}a& 1& d\\ b& 1& e\\ c& 1& f\end{array}\right]\mathbf{=}\mathbf{7}$ and role="math" localid="1660392376197" $\mathit{d}\mathit{e}\mathit{t}\mathbf{\left[}\begin{array}{ccc}\mathbf{a}& \mathbf{1}& \mathbf{d}\\ \mathbf{b}& \mathbf{2}& \mathbf{e}\\ \mathbf{c}& \mathbf{3}& \mathbf{f}\end{array}\mathbf{\right]}\mathbf{=}\mathbf{11}$,

a. Find $\mathit{d}\mathit{e}\mathit{t}\left[\begin{array}{ccc}a& 3& d\\ b& 3& e\\ c& 3& f\end{array}\right]$

b. Find$\mathit{d}\mathit{e}\mathit{t}\left[\begin{array}{ccc}a& 3& d\\ b& 5& e\\ c& 7& f\end{array}\right]$

Short answer

Therefore,

a) $\left|\begin{array}{ccc}a& 3& d\\ b& 3& 3\\ c& 3& f\end{array}\right|=21$

b) $\left|\begin{array}{ccc}a& 3& d\\ b& 5& e\\ c& 7& f\end{array}\right|=29$

Step-by-step solution

Step by Step solution: Step 1: Definition

A determinant is a unique number associated with asquare matrix.

A determinant is a scalar value that is a function of the entries of a square matrix.

It is the signed factor by which areas are scaled by this matrix. If the sign is negative the matrix reverses orientation.

Given

Given matrix,

$det\left[\begin{array}{ccc}\mathrm{a}& 1& \mathrm{d}\\ \mathrm{b}& 1& \mathrm{e}\\ \mathrm{c}& 1& \mathrm{f}\end{array}\right]=7$

$det\left[\begin{array}{ccc}\mathrm{a}& 1& \mathrm{d}\\ \mathrm{b}& 2& \mathrm{e}\\ \mathrm{c}& 3& \mathrm{f}\end{array}\right]=11$

(a)Step 3: To find det[a3db3ec3f]

To find $det\left[\begin{array}{ccc}\mathrm{a}& 3& \mathrm{d}\\ \mathrm{b}& 3& \mathrm{e}\\ \mathrm{c}& 3& \mathrm{f}\end{array}\right]$

Let,

$$\begin{gathered} \left|\begin{array}{ccc}a& 3& d\\ b& 3& 3\\ c& 3& f\end{array}\right|=3.\left|\begin{array}{ccc}a& 1& d\\ b& 1& e\\ c& 1& f\end{array}\right| \\ \left|\begin{array}{ccc}a& 3& d\\ b& 3& 3\\ c& 3& f\end{array}\right|=3.7 \\ \left|\begin{array}{ccc}a& 3& d\\ b& 3& 3\\ c& 3& f\end{array}\right|=21 \end{gathered}$$

(b)Step 4: To find det|a3db5ec7f|

To find $det\left|\begin{array}{ccc}\mathrm{a}& 3& \mathrm{d}\\ \mathrm{b}& 5& \mathrm{e}\\ \mathrm{c}& 7& \mathrm{f}\end{array}\right|$

Let,

$$\begin{gathered} \left|\begin{array}{ccc}\mathrm{a}& 3& \mathrm{d}\\ \mathrm{b}& 5& \mathrm{e}\\ \mathrm{c}& 7& \mathrm{f}\end{array}\right|=2\left|\begin{array}{ccc}\mathrm{a}& 1& \mathrm{d}\\ \mathrm{b}& 2& \mathrm{e}\\ \mathrm{c}& 3& \mathrm{f}\end{array}\right|+\left|\begin{array}{ccc}\mathrm{a}& 1& \mathrm{d}\\ \mathrm{b}& 1& \mathrm{e}\\ \mathrm{c}& 1& \mathrm{f}\end{array}\right| \\ \left|\begin{array}{ccc}\mathrm{a}& 3& \mathrm{d}\\ \mathrm{b}& 5& \mathrm{e}\\ \mathrm{c}& 7& \mathrm{f}\end{array}\right|=2.11+7 \\ \left|\begin{array}{ccc}\mathrm{a}& 3& \mathrm{d}\\ \mathrm{b}& 5& \mathrm{e}\\ \mathrm{c}& 7& \mathrm{f}\end{array}\right|=29 \end{gathered}$$

Therefore,

a) $\left|\begin{array}{ccc}a& 3& d\\ b& 3& 3\\ c& 3& f\end{array}\right|=21$

b) $\left|\begin{array}{ccc}a& 3& d\\ b& 5& e\\ c& 7& f\end{array}\right|=29$