Question:
If $2A+3B =\begin{bmatrix}
{2}&{-1} &{4}\\
{3}&{2}& {5} \\
\end{bmatrix} $ and $A+2B \begin{bmatrix}
{5}&{0} &{3}\\
{1}&{6}& {2} \\
\end{bmatrix} $then $B =$
If $2A+3B =\begin{bmatrix}
{2}&{-1} &{4}\\
{3}&{2}& {5} \\
\end{bmatrix} $ and $A+2B \begin{bmatrix}
{5}&{0} &{3}\\
{1}&{6}& {2} \\
\end{bmatrix} $then $B =$
Updated On: May 20, 2022
- $\begin{bmatrix} {8}&{1} &{2}\\ {1}&{10}& {1} \\ \end{bmatrix} $
- $\begin{bmatrix} {8}&{1} &{-2}\\ {-1}&{10}& {-1} \\ \end{bmatrix} $
- $\begin{bmatrix} {8}&{1} &{2}\\ {-1}&{10}& {-1} \\ \end{bmatrix} $
- $\begin{bmatrix} {8}&{-1} &{2}\\ {-1}&{10}& {-1} \\ \end{bmatrix} $
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The Correct Option is C
Solution and Explanation
We have
$2A + 3B = \begin{bmatrix}2&-1&4\\ 3&2&5\end{bmatrix}\,\,\,\,\, \dots(i)$
and $ A + 2B = \begin{bmatrix}5&0&3\\ 1&6&2\end{bmatrix} \,\,\,\,\,\dots(ii)$
Multiply E (ii) by 2 and subtracting E(i) from (ii), we get
$ B = 2 \begin{bmatrix}5&0&3\\ 1&6&2\end{bmatrix} -\begin{bmatrix}2&-1&4\\ 3&2&5\end{bmatrix} $
$= \begin{bmatrix}8&1&2\\ -1&10&-1\end{bmatrix}$
$2A + 3B = \begin{bmatrix}2&-1&4\\ 3&2&5\end{bmatrix}\,\,\,\,\, \dots(i)$
and $ A + 2B = \begin{bmatrix}5&0&3\\ 1&6&2\end{bmatrix} \,\,\,\,\,\dots(ii)$
Multiply E (ii) by 2 and subtracting E(i) from (ii), we get
$ B = 2 \begin{bmatrix}5&0&3\\ 1&6&2\end{bmatrix} -\begin{bmatrix}2&-1&4\\ 3&2&5\end{bmatrix} $
$= \begin{bmatrix}8&1&2\\ -1&10&-1\end{bmatrix}$
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Concepts Used:
Matrices
Matrix:
A matrix is a rectangular array of numbers, variables, symbols, or expressions that are defined for the operations like subtraction, addition, and multiplications. The size of a matrix is determined by the number of rows and columns in the matrix.
The basic operations that can be performed on matrices are:
- Addition of Matrices - The addition of matrices addition can only be possible if the number of rows and columns of both the matrices are the same.
- Subtraction of Matrices - Matrices subtraction is also possible only if the number of rows and columns of both the matrices are the same.
- Scalar Multiplication - The product of a matrix A with any number 'c' is obtained by multiplying every entry of the matrix A by c, is called scalar multiplication.
- Multiplication of Matrices - Matrices multiplication is defined only if the number of columns in the first matrix and rows in the second matrix are equal.
- Transpose of Matrices - Interchanging of rows and columns is known as the transpose of matrices.