Applications of Matrices and Determinants: One Mark Questions, Book Back Answers and Notes

About Applications of Matrices and Determinants

Medium ~180 min study

A set of linear equations such as "2 pens and 3 notebooks cost 70 rupees" turns up in physics, economics and everyday problems. When there are three or more unknowns, solving by substitution gets messy. Writing the system as a single matrix equation $AX = B$ keeps the work organised: $A$ holds the coefficients, $X$ the unknowns and $B$ the constants.

The chapter starts from what you already know about matrix multiplication and determinants. It builds the adjoint, $\text{adj}\,A$, and then the inverse $A^{-1} = \frac{1}{|A|}\,\text{adj}\,A$, which exists only when $|A| \neq 0$. Next comes the rank of a matrix, found by reducing the matrix to row-echelon form with elementary row operations. Rank is what tells you whether a system has a unique solution, infinitely many solutions or no solution. Geometrically, that is the difference between three planes meeting at one point, along a line, or not at all.

Most of the marks here come from careful procedure. One wrong sign in a cofactor or one slip in a row operation ruins the whole answer, so check $|A|$ first and verify your inverse by multiplying back to get $I$. Expect questions on testing consistency with the Rouché-Capelli theorem and on solving word problems by Gaussian elimination.

What you'll learn

• Find the adjoint and inverse of a non-singular square matrix and check the answer using $AA^{-1} = I$.
• Reduce a matrix to row-echelon form using elementary row operations.
• Find the rank of a matrix from its echelon form or from its minors.
• Solve non-homogeneous systems by the matrix inversion method, Cramer's rule and Gaussian elimination.
• Test a system for consistency with the Rouché-Capelli theorem and decide how many solutions it has.
• Use matrices to code and decode messages and to balance chemical equations.

Before you start

• Addition, subtraction and multiplication of matrices.
• Evaluating 2x2 and 3x3 determinants and finding cofactors.
• The identity matrix, the transpose and scalar multiplication.

Applications of Matrices and Determinants explained

How Matrices Solve Linear Systems

Inverse of a Non-Singular Square Matrix

A square matrix $A$ is non-singular when $|A| \neq 0$, and only then does it have an inverse. To find it, write the matrix of cofactors, take its transpose to get $\text{adj}\,A$, and divide by $|A|$. The inverse satisfies $AA^{-1} = A^{-1}A = I$. Learn the standard properties: the reversal laws $(AB)^{-1} = B^{-1}A^{-1}$ and $(A^T)^{-1} = (A^{-1})^T$, and results such as $|\text{adj}\,A| = |A|^{n-1}$ for a matrix of order $n$. These make many one-mark questions quick. Once you have $A^{-1}$, a matrix equation $AX = B$ gives $X = A^{-1}B$.

Elementary Transformations and Rank of a Matrix

There are three elementary row operations: interchange two rows, multiply a row by a non-zero number, and add a multiple of one row to another. The same three exist for columns. None of them changes the rank. Use them to bring the matrix to row-echelon form, where every zero row is at the bottom and each leading non-zero entry sits to the right of the one above it. The rank is then the number of non-zero rows. The textbook also defines rank as the order of the highest-order non-vanishing minor; both definitions give the same number.

Matrix Inversion Method and Cramer's Rule

Both methods need $|A| \neq 0$. In the matrix inversion method you compute $X = A^{-1}B$. In Cramer's rule you replace one column of $A$ at a time by the constants to get $\Delta_1, \Delta_2, \Delta_3$, and then $x = \frac{\Delta_1}{\Delta}$, $y = \frac{\Delta_2}{\Delta}$, $z = \frac{\Delta_3}{\Delta}$. Cramer's rule is quick for two or three unknowns, but the number of determinants grows fast for bigger systems.

Gaussian Elimination and the Gauss-Jordan Method

In Gaussian elimination you write the augmented matrix $[A \mid B]$, reduce it to row-echelon form, and then solve by back-substitution, starting from the last equation. The Gauss-Jordan method keeps reducing until the coefficient part becomes the identity matrix; the textbook uses it to find the inverse of a matrix by reducing $[A \mid I]$ to $[I \mid A^{-1}]$. Elimination has one big advantage over Cramer's rule: it still works when the system has no solution or infinitely many, and it shows you which case you are in.

Consistency and the Rouché-Capelli Theorem

For a system $AX = B$ in $n$ unknowns, compare $\rho(A)$ with $\rho([A \mid B])$. If they are different, the system is inconsistent and has no solution. If they are equal, the system is consistent. When the common rank equals $n$, the solution is unique. When the common rank is less than $n$, there are infinitely many solutions, described using $n - \rho(A)$ free parameters. A homogeneous system $AX = O$ is always consistent, because $X = O$ works; it has non-trivial solutions exactly when $\rho(A) < n$, which for a square matrix means $|A| = 0$.

Common mistakes to avoid

• Starting to find $A^{-1}$ without first checking $|A|$. If $|A| = 0$ there is no inverse, and all that work is wasted.
• Forgetting to transpose the cofactor matrix, so the answer is the cofactor matrix and not the adjoint.
• Making an arithmetic slip in a row operation, which gives the wrong echelon form and the wrong rank. Write down each operation, such as $R_2 \to R_2 - 2R_1$.
• Reading the Rouché-Capelli test the wrong way round. Unequal ranks mean no solution; equal ranks below $n$ mean infinitely many solutions, not no solution.
• Using Cramer's rule when $\Delta = 0$. The rule does not apply then; use the rank method or elimination instead.

Test yourself on these with the practice test, then check the worked reasoning in the solved MCQs.

Frequently asked questions

What is a non-singular matrix?

It is a square matrix whose determinant is not zero. Such a matrix has exactly one inverse. If the determinant is zero the matrix is singular and has no inverse. A system $AX = B$ with a singular $A$ is then either inconsistent or has infinitely many solutions; it never has a unique solution.

How do I find the rank of a matrix?

Reduce the matrix to row-echelon form with elementary row operations and count the non-zero rows. For a small matrix you can also look for the largest square submatrix with a non-zero determinant; its order is the rank. For example, a 3x3 matrix with $|A| \neq 0$ has rank 3.

When should I use Cramer's rule?

Use it for two or three unknowns when $\Delta$, the determinant of the coefficient matrix, is not zero. It gives each unknown directly as a ratio of determinants. For larger systems it needs too many determinants, so Gaussian elimination is quicker.

What does it mean for a system to be inconsistent?

No set of values satisfies all the equations together. For two equations in two unknowns this means two parallel lines. In rank language, the rank of $[A \mid B]$ is greater than the rank of $A$. When you reduce the augmented matrix you will see a row like $[0\ 0\ 0 \mid 5]$, which says $0 = 5$.

What is the difference between Gaussian elimination and the Gauss-Jordan method?

Gaussian elimination stops at row-echelon form, and you then use back-substitution to find the unknowns. The Gauss-Jordan method continues until the coefficient part is the identity matrix (reduced row-echelon form), so the values can be read straight off. In the textbook, Gauss-Jordan is mainly used to find the inverse of a matrix.

How are matrices used to code messages?

Letters are changed to numbers (A = 1, B = 2 and so on, with 0 for a blank) and grouped into small row matrices. Each is multiplied by a non-singular encoding matrix to give the coded message. The receiver multiplies each coded matrix by the inverse of the encoding matrix to get back the original numbers, and so the original message.