Toppers Study Toppers Study Toppers StudyDetailed NCERT Solutions for 12 Mathematics I 4. Determinants to simplify learning. Understand chapters clearly and practice with free solutions for better results.
Detailed NCERT Solutions for 12 Mathematics I 4. Determinants to simplify learning. Understand chapters clearly and practice with free solutions for better results.
Preparing for exams becomes easier with Exercise 4.5. Whether you are studying for board exams or mid-term exams, 12 Mathematics I Chapter 4. Determinants solutions provide quick revising points, well-structured answers, and additional practice material to help you score better.
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Exercise 4.5
Find adjoint of each of the matrices in Exercise 1 and 2.
Ques.1. 
Ans. Here A = 
A11 = Cofactor of 
A12 = Cofactor of 
A21 = Cofactor of 
A22 = Cofactor of 
adj. A = 
= 
Ques.2. 
Ans. Here A = 
= 
adj. A = 
Verify A (adj. A) = 
in Exercise 3 and 4.
Ques.3. 
Ans. Let A = 
adj. A = 
A.(adj. A) =
= 
= 
…..(i)
Again (adj. A). A =
= 
= …..(ii)
And 
= 
Again 
…..(iii)
From eq. (i), (ii) and (iii)
A. (adj. A) = (adj. A). A =
Ques.4. 
Ans. Let A = 
= 
adj. A = 
A. (adj. A) = 
= 
= 
……….(i)
Again (adj. A). A =
= 
= ……….(ii)
And
= 
Also 
= ……….(iii)
From eq. (i), (ii) and (iii) A. (adj. A) = (adj. A). A =
Ques.5. 
Ans. Let A = 
= 
0
Matrix A is non-singular and hence 
exist.
Now adj. A = 
And 
Ques.6. 
Ans. Let A =
= 
Matrix A is non-singular and hence exist.
Now adj. A = 
And 
Ques.7. 
Ans. Let A = 
= 
exists.
A11 = 
, A12 = 
,
A13 = 
, A21 = 
,
A22 = 
, A23 = 
,
A31 = 
, A32 = 
,
A33 = 
adj. A = 
Ques.8. 
Ans. Let A = 
= 
exists.
A11 = 
, A12 = 
,
A13 = 
, A21 = 
,
A22 = 
, A23 = 
,
A31 = 
, A32 = 
,
A33 = 
adj. A = 
Ques.9. 
Ans. Let A = 
= 
exists.
A11 = 
, A12 = 
,
A13 = 
, A21 = 
,
A22 = 
, A23 = 
,
A31 = 
, A32 = 
,
A33 = 
adj. A = 
Ques.10. 
Ans. Let A = 
= 
exists.
A11 = 
, A12 = 
,
A13 = 
, A21 = 
,
A22 = 
, A23 = 
,
A31 = 
, A32 = 
,
A33 = 
adj. A = 
Ques.11. 
Ans. Let A = 
= 
exists.
A11 = 
,
A12 = 
, A13 = 
,
A21 = 
, A22 = 
,
A23 = 
, A31 = 
,
A32 = 
, A33 = 
adj. A = 
and B = 
verify that 
Ans. Given: Matrix A =
= 15 – 14 = 1 
0
= 
Matrix B = 
= 54 – 56 = 
0
Now AB = = 
= 
= 
Now L.H.S. = 
……….(i)
R.H.S. = 
= 
= 
……….(ii)
From eq. (i) and (ii), we get
L.H.S. = R.H.S.
, show that A2 – 5A + 7I = 0. Hence find 
Ans. Given: A = 
L.H.S. = 
= 
= 
= 
= 
= 
= R.H.S.
……(i)
To find: , multiplying eq. (i) by .
= 
= 
= 
Ques.14. For the matrix A = 
find numbers 
and 
such that 
Ans. Given: A = 
We have 
……….(i)
Here satisfies 
also, so
Putting in eq. (i), 
Here also satisfies 
, so
hance, and
, show that 
Hence find Ans. Given: A = 
= 
Now 
= 
= 
L.H.S. = 
= 
= 
= 
= 
= 
= R.H.S.
No
Exercise 4.5 are created by experts to give step-by-step explanations. Around 60–70% of exam questions are based on NCERT concepts. Our 12 Mathematics I Chapter 4. Determinants solutions help you understand the core concepts and practice effectively.
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Every NCERT chapter ends with exercises, and solving them is crucial. Our Exercise 4.5 include complete solutions for 12 Mathematics I Chapter 4. Determinants exercises. With step-by-step answers, you gain clarity and confidence to attempt similar exam questions.
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Our Exercise 4.5 are useful for both board exams and mid-term exams. For 12 Mathematics I, we provide notes, exercises, and important Q&A so that you can revise smartly and write perfect answers in exams.
In short, Exercise 4.5 for 12 Mathematics I Chapter 4. Determinants are a complete study package. With quick revising points, NCERT exercises, and additional important questions, you can prepare effectively for exams. Make these solutions your study companion and excel in your academic journey.
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