Tag: terms related to matrices

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Questions Related to terms related to matrices

Multiple choice terms related to matrices matrices and determinants matrices algebra maths

Let $A+2B=\begin{bmatrix} 1 & 2 & 0 \ 6 & -3 & 3 \ -5 & 3 & 1 \end{bmatrix}$ and $2A-B=\begin{bmatrix} 2 & -1 & 5 \ 2 & -1 & 6 \ 0 & 1 & 2 \end{bmatrix}$, then $tr(A)-tr(B)$ has the value equal to

  1. 0

  2. 1

  3. 2

  4. none of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$A+2B=\begin{bmatrix} 1 & 2 & 0 \ 6 & -3 & 3 \ -5 & 3 & 1 \end{bmatrix}$         .....(i)

$2A-B=\begin{bmatrix} 2 & -1 & 5 \ 2 & -1 & 6 \ 0 & 1 & 2 \end{bmatrix}$         .....(ii)

$\Rightarrow 4A-2B=\begin{bmatrix} 4 & -2 & 10 \ 4 & -2 & 12 \ 0 & 2 & 4 \end{bmatrix}$      ....(iii)

Adding (i) and (iii), we get
$5A=\begin{bmatrix} 5 & 0 & 10 \ 10 & -5 & 15 \ -5 & 5 & 5 \end{bmatrix}$

$\Rightarrow A=\begin{bmatrix} 1 & 0 & 2 \ 2 & -1 & 3 \ -1 & 1 & 1 \end{bmatrix}$
So, $tr(A)=1$

Now, by eq(ii),
$B=\begin{bmatrix} 2 & 0 & 4 \ 4 & -2 & 6 \ -2 & 2 & 2 \end{bmatrix}-\begin{bmatrix} 2 & -1 & 5 \ 2 & -1 & 6 \ 0 & 1 & 2 \end{bmatrix}$

$\Rightarrow B=\begin{bmatrix} 0 & 1 & -1 \ 2 & -1 & 0 \ -2 & 1 & 0 \end{bmatrix}$
So, $tr(B)=-1$
Now, $tr(A)-tr(B)=1+1=2$

Multiple choice terms related to matrices matrices and determinants matrices algebra maths

Let $A$ be the $2\times2$ matrices given by $A=\left[a _{ij}\right]$ where $a _{ij} = \left{0,1,2,3,4\right}$ such that $a _{11} + a _{12} + a _{21} + a _{22} = 4$
Find the number of matrices $A$ such that the trace of $A$ is equal to 4

  1. 3

  2. 4

  3. 5

  4. 6
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given $tr(A)=4$
$\Rightarrow a _{11}+a _{22}=4$

$a _{ ij }={ { 0,1,2,3,4}  }$

So, diagonal entries of A can be 0 and 4 , 4 and 0, 1 and 3, 3 and 1, 2 and 2,
Hence, 5 matrices are possible

Multiple choice terms related to matrices matrices and determinants matrices algebra maths

If $A=\begin{bmatrix} 2 & 1 \ 4 & 1 \end{bmatrix}; B=\begin{bmatrix} 3 & 4 \ 2 & 3 \end{bmatrix}$ and $C=\begin{bmatrix} 3 & -4 \ -2 & 3 \end{bmatrix}$ then $tr(A)+tr\left( \dfrac { ABC }{ 2 }  \right) +tr\left( \dfrac { A{ \left( BC \right)  }^{ 2 } }{ 4 }  \right) +tr\left( \dfrac { A{ \left( BC \right)  }^{ 3 } }{ 8 }  \right) +......\infty $ =

  1. $6$
  2. $9$
  3. $12$
  4. $15$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The expression is a geometric series of traces. Given the matrices, the sum converges to a specific value based on the properties of the trace and matrix multiplication.

Multiple choice terms related to matrices matrices and determinants matrices algebra maths

Consider three matrices A= $ \begin{bmatrix} 2 & 1 \ 4 & 1 \end{bmatrix} $, $ B = \begin{bmatrix} 3 & 4 \ 2 & 3 \end{bmatrix} $ and $ C = \begin{bmatrix} 3 & -4 \ -2 & 3 \end{bmatrix} $ Then the value of the sum 
$ tr(A)+tr \cfrac {(ABC) } {2} +tr \cfrac {A( {BC})^2} {4}+ \cfrac {A( {BC})^3} {2}  +...+ \infty               $is

  1. 6

  2. 9

  3. 12

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The matrix product BC results in the identity matrix, meaning powers of BC remain the identity matrix. Summing the resulting traces forms a geometric series whose sum evaluates precisely to 6.

Multiple choice terms related to matrices matrices and determinants matrices algebra maths

 $P=\left[ \begin{matrix} { 5a }^{ 2 }+2bc & 6 & 8 \ 13 & { 8b }^{ 2 }-10ac & -9 \ -7 & 5 & { 25c }^{ 2 } \end{matrix} \right]$ and $Q=\left[ \begin{matrix} { a }^{ 2 }+6bc & 3 & 5 \ 12 & { -b }^{ 2 } & 6 \ 1 & 4 & { 17bc }^{ 2 } \end{matrix} \right] a,b$ & $c \epsilon N$, if trace $\left(P\right)=trac\left(Q\right)$, and $a,b$ & $C$ are sides of $\Delta ABC$ with $BC=a,CA=b$ & $AB=C$ then $\cos A$ is:

  1. $\dfrac{-79}{120}$
  2. $\dfrac{-89}{120}$
  3. $\dfrac{-33}{40}$
  4. $\dfrac{-31}{40}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Equating the traces of P and Q gives an equation involving a, b, and c. Using the properties of triangle sides and the Law of Cosines, one can solve for cos A.

Multiple choice terms related to matrices matrices and determinants matrices algebra maths

If $\left( \begin{array} { l l } { 3 } & { 2 } \ { 7 } & { 5 } \end{array} \right) A \left( \begin{array} { c c } { - 1 } & { 1 } \ { - 2 } & { 1 } \end{array} \right) = \left( \begin{array} { c c } { 2 } & { - 1 } \ { 0 } & { 4 } \end{array} \right)$  then trace of  $A$  is equal to

  1. $-25$
  2. $-21$
  3. $-15$
  4. $-11$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let M1 * A * M2 = M3. Then A = M1^-1 * M3 * M2^-1. Calculate the inverse of the matrices and perform the multiplication to find A, then sum the diagonal elements.