2000017610
Level:
C
For which positive real number \(a\) is the determinant of the matrix equal to \(36\)?
\[
\left (\array{
2a & 0 & a\cr
0 & 1& -1\cr
a & 2a & 6\cr
} \right )
\]
\( a=2\)
\( a=4\)
\( a=6\)
\(a=0\)
Matrices and determinants
2000018401
Level:
C
Assuming that
\[
\left|
\array{
1 & 1 & 1\cr
a & b& c\cr
x & y & z\cr
}
\right| =5
\]
compute the following determinant:
\[
\left|
\array{
a & b & c\cr
x & y& z\cr
3 & 3 & 3\cr
}
\right|.
\]
\(15\)
\(-15\)
\(27\)
other
2000018402
Level:
C
Which two of the matrices, \(A\), \(B\), and \(C\) have the same determinant?
\[
A=\left(
\array{
1 & 3& 5\cr
5 & 3& 2\cr
1 & 5 & 3\cr
}
\right),~
B=\left(
\array{
1 & 2& 5\cr
5 & 3& 3\cr
1 & 5 & 3\cr
}
\right),~
C=\left(
\array{
1 & 5& 1\cr
3 & 3& 5\cr
5 & 2 & 3\cr
}
\right)
\]
\(A\) and \(C\)
\(A\) and \(B\)
\(B\) and \(C\)
none
2000018403
Level:
C
Assuming that
\[
\left|
\array{
1 & 1& 1\cr
a & b& c\cr
x & y & z\cr
}
\right|=5
\]
compute the following determinant.
\[
\left|
\array{
5 & 5& 5\cr
a+2 & b+2& c+2\cr
\frac{x}3 & \frac{y}3 & \frac{z}3\cr
}
\right|
\]
\(\frac{25}3\)
\(\frac{5}3\)
\(7\)
\(\frac{7}3\)
2000018404
Level:
C
Assuming that
\[
\left|
\array{
4 & 0& 2\cr
a & b& c\cr
2 & 4 & 6\cr
}
\right|=36
\]
compute the following determinant.
\[
\left|
\array{
1 & 2& 3\cr
a & b& c\cr
4 & 0 & 2\cr
}
\right|
\]
\(-18\)
\(18\)
\(72\)
\(-72\)
2000018405
Level:
C
Assuming that
\[
\left|
\array{
1 & a& 5\cr
1 & b& 6\cr
1 & c & 7\cr
}
\right|=-4
\]
compute the following determinant:
\[
\left|
\array{
a & -2& 6\cr
b & -2& 7\cr
c & -2 & 8\cr
}
\right|
\]
\(-8\)
\(8\)
\(-7\)
\(9\)
2000018406
Level:
C
Which matrix from \(A\), \(B\), \(C\) and \(D\) does not have a determinant equal to zero?
\[\]
$A=\left(
\array{
1 & 2& 5\cr
1 & 3& 6\cr
1 & 4 & 7\cr
}
\right),$
$B=\left(
\array{
1 & 2& 3\cr
0 & 1& -1\cr
2 & 4 & 6\cr
}
\right),$
$C=\left(
\array{
1 & 1& 1\cr
2 & 3& 4\cr
15 & 16 & 17\cr
}
\right),$
$D=\left(
\array{
1 & 2& 5\cr
1 & -4& -6\cr
1 & -4 & 7\cr
}
\right)$
\(D\)
\(A\)
\(B\)
\( C\)
2000018407
Level:
C
Compute the determinant of the following matrix:
\[
A=\left(
\array{
1 & 0& 0\cr
2 & 3& 0\cr
-1 & 4 & 11\cr
}
\right)
\]
\(33\)
\(0\)
\(34\)
\( 15\)