Use the appropriate property of determinants to find a. Do not evaluate the determinants. vert } 4&23&-3 -43&-31&-17 3&-3&9 vert Answer: a=

To find the value of \( a \), we can use the property of determinants that states that multiplying a row (or column) of a matrix by a scalar multiplies the determinant of the matrix by the same scalar.<br /><br />In this case, we have the determinant of a 3x3 matrix and we want to find the value of \( a \) such that the determinant of the matrix remains unchanged when the first row is multiplied by \( a \).<br /><br />The given determinant is:<br />\[ \left| \begin{matrix} 4 & 23 & -3 \\ -43 & -31 & -17 \\ 3 & -3 & 9 \end{matrix} \right| \]<br /><br />We want to find \( a \) such that:<br />\[ \left| \begin{matrix} 4 & 23 & -3 \\ -43 & -31 & -17 \\ 3 & -3 & 9 \end{matrix} \right| = a \cdot \left| \begin{matrix} -4 & -23 & 3 \\ 43 & 31 & 17 \\ -3 & 3 & -9 \end{matrix} \right| \]<br /><br />Comparing the two determinants, we can see that the second determinant is obtained by multiplying the first row of the original determinant by \( -a \). Therefore, we have:<br />\[ -a \cdot \left| \begin{matrix} 4 & 23 & -3 \\ -43 & -31 & -17 \\ 3 & -3 & 9 \end{matrix} \right| = a \cdot \left| \begin{matrix} -4 & -23 & 3 \\ 43 & 31 & 17 \\ -3 & 3 & -9 \end{matrix} \right| \]<br /><br />Since the determinants are equal, we have:<br />\[ -a = a \]<br /><br />Solving for \( a \), we get:<br />\[ 2a = 0 \]<br />\[ a = 0 \]<br /><br />Therefore, the value of \( a \) is \( 0 \).