If D=| 3 ! 4 ! 5 ! 4 ! 5 ! 6 ! 5 ! 6 ! 7 ! | then the la

If D=| 3 ! 4 ! 5 ! 4 ! 5 ! 6 ! ...

Key Concepts

Matrix Determinant

A matrix determinant is a scalar value derived from a square matrix that encapsulates many important properties of the matrix. It is used to determine if the matrix is invertible, can help solve systems of linear equations, and can be computed using various methods such as cofactor expansion. In problems involving determinants, manipulations such as factor extraction and row reductions can simplify calculations significantly.

Factorial

A factorial, denoted by n!, is the product of all positive integers up to the number n. Factorials grow very rapidly and have properties that allow them to be factored out or cancelled in algebraic expressions, including determinants. Recognizing and using these properties can simplify complex expressions involving factorials.

Modular Arithmetic

Modular arithmetic involves calculations with residues after division by a fixed modulus, providing a way to work with remainders. This is especially useful when determining the last digit of a large number or expression, as it reduces the arithmetic to a manageable form by focusing only on the units digit (or another digit specified by the modulus).

Cyclic Patterns in Exponentiation

Exponentiation modulo a number often exhibits a cyclic pattern, meaning that the residues repeat after some period. Recognizing these cycles allows one to reduce the exponent by the length of the cycle, making it feasible to compute large exponents modulo a given number such as 10 to find the last digit.

Transcript

00:01 So in the given question, we are told that there is a determinant, there is a matrix d which is given as 3 factorial, 4 factorial, 5 factorial, 4 factorial, 5 factorial, 6 factorial, 5 factorial, 5 factorial, 6 factorial, 5 factorial, 6 factorial, 7 factorial, and we are told to find the last digit if d is equal to this the value of this determinant then we are told to find the value of the last digit of the expression d divided by 216 minus 4 raised to the power 401 so this the last digit of this number is what we need to find.

01:05 So what can we start with? we can start with the matrix with the determinant that is given and we can take 3 factorial we can take 3 factorial 4 factorial and 5 factorial as a common factor from this determinant.

01:26 So since we can write 3 factorial 4 factorial as 4 times 3 factorial and 5 factor as 5 times 4 times 3 factorial we have 3 factorial as a common factor in the first row similarly we will have 4 factorial as a common factor in the third second row and 5 factorial as a common factor in the third row right so now since we have common factors in each row let's see take them outside so 3 factorial times 4 factorial times 5 factorial and inside we would have 1 420 1 1 5 30 and 1 642 right and in the next step what we are going to do is we can write we can do the row operation r2 changes to r2 minus r1 and r3 changes to r2 minus r1 and r3 changes to r3 minus r1 2.

02:50 So when we do this what we get is we will get 3 factorial times 4 factorial times 5 factorial.

03:00 The first row is not doesn't have any change and in the second row we have one minus 1 which is 0 5 minus 4 is equal to 1 30 minus 20 is 10 again in the third row we have 1 minus 1 which is 0 6 minus 4 is 2 and 42 minus 20 is 22 right and now we can find the determinant of this so we will have 3 factorial times 4 factorial times 5 factorial times 1 times 22 minus 20 and this would give us 2 as the answers.

03:48 So 3 factorial times 4 factorial times 5 factorial times 22 minus 2 is 2.

03:55 So this is what we get as d.

03:58 Now, let's find what is, let's take the expression d and when we multiply, when we take d divided by 216 minus 4, what we would get is let's evaluate d first...

Please give Ace some feedback