The value of the determinant| beginmatrix1&1&1 mC1&m+1C1&m+2C1 mC2&m+1C2&m+2C2 endmatrix|is equal to

Question

The value of the determinant| beginmatrix1&1&1 mC1&m+1C1&m+2C1 mC2&m+1C2&m+2C2 endmatrix|is equal to (A) 1 (B)  -1 (C) 0 (D) none of these

Tardigrade Admin · Accepted Answer

| beginmatrix1&1&1 mC1& m+1C1& m+2C1 mC2& m+1C2& m+2C2 endmatrix| =| beginmatrix1&1&1 mC1& m+1C1& m+1C0+ m+1C1 mC2& m+1C2& m+1C1+ m+1C2 endmatrix| =| beginmatrix1&1&0 mC1& m+1C1& m+1C0 mC2& m+1C2& m+1C1 endmatrix| [ textApplying C3→ C3-C2] =| beginmatrix1&1&0 mC1& mC0+ mC1& m+1C0 mC2& mC1+ mC2&m+1C1 endmatrix| =| beginmatrix1&0&0 mC1& mC0& m+1C0 mC2& mC1& m+1C1 endmatrix| [ textApplying C2→ C2-C1] = mC0 m+1C1- m+1C0 mC1=m+1-m=1

Q. The value of the determinant $∣ ∣ 1^{m} C_{1}^{m} C_{2} 1^{m + 1} C_{1}^{m + 1} C_{2} 1^{m + 2} C_{1}^{m + 2} C_{2} ∣ ∣$ is equal to

$1$

$- 1$

$0$

none of these

Q. The value of the determinant∣∣​1mC1​mC2​​1m+1C1​m+1C2​​1m+2C1​m+2C2​​∣∣​is equal to

1

−1

0

none of these

Q. The value of the determinant $∣ ∣ 1^{m} C_{1}^{m} C_{2} 1^{m + 1} C_{1}^{m + 1} C_{2} 1^{m + 2} C_{1}^{m + 2} C_{2} ∣ ∣$ is equal to

$1$

$- 1$

$0$