# Simultaneous Algebraic Equations Questions and Answers – Introduction to Determinants

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This set of Numerical Methods Multiple Choice Questions & Answers (MCQs) focuses on “Introduction to Determinants”.

1. For a determinant containing 4 elements a1 b1 a2 b2, what will the elements of the leading diagonal?
a) a1 b1
b) a1 b2
c) a2 b1
d) a2 b2

Explanation: The diagonal having its first element in the top left corner of the matrix is considered as the leading diagonal of the determinant. And here, it starts from a1 and ends at b2.

2. For a determinant $$\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}$$, the cofactor of h will be equal to which of the following?
a) (-1)3+2 $$\begin{vmatrix}a&c\\d&f\end{vmatrix}$$
b) (-1)3+2 $$\begin{vmatrix}d&f\\g&i\end{vmatrix}$$
c) (-1)3+2 $$\begin{vmatrix}a&c\\g&i\end{vmatrix}$$
d) (-1)3+2 $$\begin{vmatrix}b&c\\e&f\end{vmatrix}$$

Explanation: The cofactor of an element is obtained by deleting the row and the column which intersect at that element with proper sign. The sign is given by (-1)i+j for ith row and jth column.

3. What is the value of the determinant $$\begin{pmatrix}2&5&1\\7&4&5\\1&2&3\end{pmatrix}$$?
a) -56
b) -96
c) -66
d) -85

Explanation: $$\begin{pmatrix}2&5&1\\7&4&5\\1&2&3\end{pmatrix}$$ = 2 $$\begin{vmatrix}4&5\\2&3\end{vmatrix}$$ -5 $$\begin{vmatrix}7&5\\1&3\end{vmatrix}$$ + 1 $$\begin{vmatrix}7&4\\1&2\end{vmatrix}$$
= 2(12-10) – 5(21-5) + 1(14-4)
= -66.

4. A determinant vanishes if one of the entire row or column consists of all zero elements?
a) True
b) False

Explanation: A determinant vanishes when two of the rows or columns have zero elements. It even vanishes when the elements of two rows or columns have identical elements.

5. If each element of a line consists of m terms, the determinant can be expressed as the sum of ______ determinants.
a) m-1
b) m
c) m+1
d) m2

Explanation: This is because while solving for a determinant, we choose a particular row or column and multiply its members with their cofactor. And, in doing so, the numbers of determinants formed are equal to the order of the matrix.

6. Consider the set P of all determinants of order 3 with entries 0 or 1 only. Let Q the subset of P consisting of all determinants with value 1.
Let R be the subset of P consisting of all determinants with value – 1. Then, which of the following is true?
a) R is empty
b) Q has as many elements as R
c) P = Q ∪ R
d) Q has twice as many elements as R

Explanation: Number of determinants in P = 29
Similarly, we found that the number of determinants with value 1 are like $$\begin{bmatrix}1&0&1\\1&1&0\\0&1&0\end{bmatrix}$$ are equal to the number of determinants with value -1 $$\begin{bmatrix}1&1&1\\1&0&0\\0&0&1\end{bmatrix}$$.

7. A matrix B and _____ will have the same determinant.
a) Its transpose
b) Its inverse
c) Its echelon matrix

Explanation: This is because a matrix and its transpose have same elements, only their position differs. The ith index changes to jth and vice versa.

8. Which of the following are true?
a) If we multiply a row in a matrix A by a real number r, |A| changes in r.|A|
b) If we multiply a column and a row in A, both by a real number r, |A| changes in r.|A|
c) If we swap two columns in A, |A| does not change sign
d) If we divide a matrix with a scalar, division occurs only in a single row or column

Explanation: Since multiplication of a matrix with a scalar is possible, so whenever we multiply a row in a matrix by a real number, it gets multiplied in the way it gets multiplied with a number.

9. If A and B are two square matrices, then |A.B| is same as which of the following?
a) |A|.|B|
b) |B|.|A|
c) |B-A|
d) |A-B|

Explanation: Determinant of a product of two square matrices is equal to product of the individual determinants of the square matrices. Matrix multiplication follows commutative property.

10. For a matrix A having |A| = 0, which of the following are true?
a) A-1 exists
b) A-1 does not exists
c) A is non-singular
d) A-1 exists with |A-1|= 0 