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# $LetA=[ac bd ]$, where $a,b,c,d∈I$ ,$(A−αI)$ is invertible for all $α∈R$, if $a=1,d=2$ then

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## Text solutionVerified

As $A−αI$ is invertible for all $α∈R$ .

$det(A−αI)=0∀α∈R$ .

$⇒(a−α)(d−α)−bc=0∀α∈R$ .

$⇒α_{2}−(a+d)α+ad−bc=0∀α∈R.$

Therefore

$(a+d)_{2}−4(ad−bc)<0⇒(a−d)_{2}+4bc<0$

Therefore, $bc<0$

Also, $a_{2}+d_{2}−2ad+4bc<0$

$⇒0≤a_{2}+d_{2}<2ad−4bc$

$⇒bc<21 ad$ .

Thus, $bc<min(0,21 ad)$

$det(A−αI)=0∀α∈R$ .

$⇒(a−α)(d−α)−bc=0∀α∈R$ .

$⇒α_{2}−(a+d)α+ad−bc=0∀α∈R.$

Therefore

$(a+d)_{2}−4(ad−bc)<0⇒(a−d)_{2}+4bc<0$

Therefore, $bc<0$

Also, $a_{2}+d_{2}−2ad+4bc<0$

$⇒0≤a_{2}+d_{2}<2ad−4bc$

$⇒bc<21 ad$ .

Thus, $bc<min(0,21 ad)$

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**LIVE**classesQuestion Text | $LetA=[ac bd ]$, where $a,b,c,d∈I$ ,$(A−αI)$ is invertible for all $α∈R$, if $a=1,d=2$ then |

Updated On | Mar 3, 2022 |

Topic | Matrices |

Subject | Mathematics |

Class | Class 12 |

Answer Type | Text solution:1 Video solution: 1 |

Upvotes | 135 |

Avg. Video Duration | 14 min |