For what value of x the matrix A is singular \[A=\left[ \begin{matrix} 1+x & 7 \\ 3-x & 8 \\ \end{matrix} \right]\]
ANSWER
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Hint: First of all try to recollect what singular matrix is and all the conditions for it. Now, find the determinant of the given \[2\times 2\] matrix and equate it to 0 to find the required value of x.
Complete step-by-step answer: Here, we have to find the value of x such that the matrix \[A=\left[ \begin{matrix} 1+x & 7 \\ 3-x & 8 \\ \end{matrix} \right]\] is singular. Before proceeding with the question, let’s see a few terms. Singular Matrix: A singular matrix refers to a matrix whose determinant is zero. Also, these matrices have no inverse. Such matrices cannot be multiplied with other matrices to achieve the identity matrix. The determinant of Matrix: For a square matrix, i.e. a matrix with the same number of rows and columns, one can capture important information about the matrix in a single number called the determinant. Now, let us consider our question. As we know that for a matrix to be singular, its determinant must be zero. So now, we find the determinant of matrix A. \[A=\left[ \begin{matrix} 1+x & 7 \\ 3-x & 8 \\ \end{matrix} \right]\] Since, we know that the determinant of any \[2\times 2\] matrix \[\left[ \begin{matrix} a & b \\ d & c \\ \end{matrix} \right]\] is given by ac – bd. So by substituting a = (1 + x), b = 7, c = 8 and d = (3 – x), we get, The determinant of matrix A \[=\left( 1+x \right)8-\left( 3-x \right)7\] Now since this matrix is singular, we will equate the above equation to zero, we get, \[\left( 1+x \right)8-7\left( 3-x \right)=0\] By simplifying the above equation, we get, \[8x+8-21+7x=0\] \[15x-13=0\] \[x=\dfrac{13}{15}\] So, we get the value of x as \[\dfrac{13}{15}\] for which matrix A would be singular.
Note: Here, students must note that we can find the determinant for the square matrix only. Also, students often make the mistake while calculating the determinant of the \[2\times 2\] matrix by taking the wrong values of a, b, c, and d. So, this mistake must be avoided. Basically, to find the determinant of \[2\times 2\] matrix, diagonal elements are multiplied, and then their difference is taken.