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Matrices and Determinants
Foundations: Structure and Operations
Matrices are rectangular arrays of numbers organized into rows and columns. Their order or dimension ($m \times n$) determines how they interact with other matrices through addition and scalar multiplication.
$$A + B = [a_{ij} + b_{ij}]$$
Matrix addition is performed element-wise by summing entries at the same index.
$a_{ij}$=Element in row $i$, column $j$(scalar)
$b_{ij}$=Corresponding element in matrix $B$(scalar)
Different Orders
→Addition is undefined; matrices must be of the same dimension.
Element Indexing: $a_{ij}$ refers to the entry in the $i^{th}$ row and $j^{th}$ column.
Dimension Rule: Addition and subtraction require exactly matching $m \times n$ dimensions.
Commutativity: Matrix addition is commutative ($A + B = B + A$), but multiplication is NOT.
Additive Identity: The Null Matrix $O$ satisfies $A + O = A$ for any matrix of the same order.
Scalar Scaling: $k(A + B) = kA + kB$ and $(k + l)A = kA + lA$ — scalar multiplication distributes over both matrix and scalar addition.
Matrix Multiplication: The Row-by-Column Rule
Unlike addition, matrix multiplication is not element-wise. It involves the dot product of rows from the first matrix and columns from the second. The inner dimensions must match for the product to be conformable.
$$c_{ij} = \sum_{k=1}^{n} a_{ik}b_{kj}$$
The result element $c_{ij}$ is the sum of products of row $i$ from $A$ and column $j$ from $B$.
$a_{ik}$=Elements of the $i^{th}$ row of matrix $A$(unitless)
$b_{kj}$=Elements of the $j^{th}$ column of matrix $B$(unitless)
$n$=Number of columns in $A$ (must equal rows in $B$)(integer)
$AB \neq BA$
→Matrix multiplication is generally non-commutative.
Dimension Matching: To multiply $(m \times n)$ by $(n \times p)$, the inner $n$ values must be equal.
Resulting Order: The product matrix will have dimensions $(m \times p)$.
Identity Property: Multiplication by the Identity Matrix ($I$) leaves the original matrix unchanged.
Determinants and Singularity
The determinant is a scalar value that captures essential properties of a square matrix. It determines if a matrix is invertible; a matrix with a determinant of zero is called singular and has no inverse.
$$|A| = a(ei - fh) - b(di - fg) + c(dh - eg)$$
Formula for a 3x3 matrix determinant via expansion by the first row.
$|A|$=Determinant of matrix $A$(scalar)
$(ei-fh)$=Minor of element $a$(scalar)
$|A| = 0$
→The matrix is singular and cannot be inverted.
2×2 Formula: For $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the determinant is simply $ad - bc$ (main diagonal minus secondary diagonal).
Cofactor Sign Pattern: Expansion uses the checkerboard $(-1)^{i+j}$, starting with $+$ at position $(1,1)$.
Product Rule: The determinant of a product equals the product of the determinants: $|AB| = |A||B|$.
Expansion Choice: You may expand along any row or column — the result is always the same. Choose the row/column with the most zeros to save work.
Properties of Determinants
Several key properties of determinants allow evaluation without full expansion. These shortcuts reduce computation by identifying special structures in the matrix.
Transpose Property: $|A| = |A^T|$. The determinant is unchanged when rows and columns are swapped.
Row/Column Interchange: Swapping two rows (or columns) reverses the sign of the determinant.
Identical Rows/Columns: If any two rows (or columns) are identical, $|A| = 0$.
Zero Row/Column: If an entire row or column is zero, $|A| = 0$.
Scalar Factor: A common factor $k$ from any single row can be taken outside: $|kR_i| = k|A|$. For the entire matrix: $|kA| = k^n|A|$.
Row Addition: Adding a scalar multiple of one row to another does not change the determinant.
Triangular Matrix: The determinant of a triangular (or diagonal) matrix equals the product of its diagonal entries.
Special Types of Matrices
Matrices are classified by the relationships between their entries. A Symmetric Matrix satisfies $A^T = A$, while a Skew-Symmetric Matrix satisfies $A^T = -A$. Similarly, a Hermitian Matrix satisfies $A^\theta = A$ for complex entries.
Symmetric: $a_{ij} = a_{ji}$ for all entries. All diagonal matrices are symmetric.
Skew-Symmetric: $a_{ij} = -a_{ji}$, which forces all diagonal entries to be zero.
Hermitian: Like symmetric, but for complex matrices — $\bar{a}_{ji} = a_{ij}$.
Skew-Hermitian: $A^\theta = -A$; diagonal entries are either zero or purely imaginary.
Null Matrix: Every entry is zero. Also called the zero matrix $O$.
Scalar Matrix: A diagonal matrix where all diagonal entries are the same scalar $k$. Equivalent to $kI$.
The Adjoint and Inverse Matrix
The inverse of a matrix $A$ is another matrix $A^{-1}$ such that their product is the Identity Matrix. Finding the inverse requires the Adjoint matrix, which is the transpose of the cofactor matrix.
$$A^{-1} = \frac{1}{|A|} \text{adj}(A)$$
The inverse is the adjoint matrix scaled by the reciprocal of the determinant.
$A^{-1}$=Multiplicative Inverse Matrix(matrix)
$\text{adj}(A)$=Adjoint matrix (Transpose of Cofactors)(matrix)
$|A|$=Determinant of A (must be non-zero)(scalar)
$|A| = 0$
→The matrix is singular and no inverse exists.
Existence: $A^{-1}$ exists if and only if $|A| \neq 0$.
Inverse Determinant: $|A^{-1}| = \frac{1}{|A|}$, since $|AA^{-1}| = |I| = 1$.
2×2 Adjoint Shortcut: For $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the adjoint is $\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$ — swap diagonals, negate off-diagonals.
Verification: Always check your answer: $AA^{-1}$ must equal $I$.
Application Strategy
Worked Examples
Problem Statement
Calculate the product $AB$ where $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$.
Matrix MultiplicationDot Product
Strategic Toolkit
$c_{ij} = \sum a_{ik}b_{kj}$
Step 1
Dimension Check
Matrix $A$ is $2 \times 2$ and $B$ is $2 \times 2$. Since inner dimensions (2 and 2) match, multiplication is possible. The result will be $2 \times 2$.
Result Dimension: $(2 \times \mathbf{2}) \cdot (\mathbf{2} \times 2) \rightarrow 2 \times 2$
Step 2
Calculate Row 1 Elements
Multiply the first row of $A$ by the columns of $B$.
$c_{11} = (1 \times 5) + (2 \times 7) = 5 + 14 = 19$
$c_{12} = (1 \times 6) + (2 \times 8) = 6 + 16 = 22$
Step 3
Calculate Row 2 Elements
Multiply the second row of $A$ by the columns of $B$.
$c_{21} = (3 \times 5) + (4 \times 7) = 15 + 28 = 43$
$c_{22} = (3 \times 6) + (4 \times 8) = 18 + 32 = 50$
Step 4
Assemble Final Matrix
Place the calculated elements into their respective positions.
$AB = \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}$
Problem Statement
Find the multiplicative inverse $A^{-1}$ of matrix $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$.
DeterminantAdjointInverse Formula
Strategic Toolkit
$A^{-1} = \frac{1}{|A|} \text{adj}(A)$
Step 1
Calculate Determinant
Find $|A|$ to check if the matrix is non-singular.
$|A| = (1 \times 4) - (2 \times 3) = 4 - 6 = -2$
Step 2
Verify Invertibility
Since $|A| = -2$ (not zero), the matrix is non-singular and the inverse exists.
$|A| \neq 0$
Step 3
Find Adjoint
For a 2x2 matrix, swap the diagonal elements and negate the off-diagonal elements.
$\text{adj}(A) = \begin{bmatrix} 4 & -2 \\ -3 & 1 \end{bmatrix}$
Step 4
Apply Formula
Scale the adjoint matrix by $1/|A|$.
$A^{-1} = \frac{1}{-2} \begin{bmatrix} 4 & -2 \\ -3 & 1 \end{bmatrix} = \begin{bmatrix} -2 & 1 \\ 1.5 & -0.5 \end{bmatrix}$
Problem Statement
Evaluate the determinant of $A = \begin{bmatrix} 2 & 3 & 1 \\ 0 & -1 & 2 \\ 1 & 4 & -1 \end{bmatrix}$.
DeterminantCofactorMinorExpansion
Strategic Toolkit
$|A| = a_{11}A_{11} + a_{12}A_{12} + a_{13}A_{13}$
Step 1
Choose Expansion Row
Expand along the first row (or any row/column — the result is the same). Use cofactors $A_{1j} = (-1)^{1+j} M_{1j}$.
$|A| = 2 \cdot A_{11} + 3 \cdot A_{12} + 1 \cdot A_{13}$
Step 2
Compute Minors
Delete row 1 and the relevant column to get each 2×2 minor.
$M_{11} = \begin{vmatrix} -1 & 2 \\ 4 & -1 \end{vmatrix} = 1 - 8 = -7$
$M_{12} = \begin{vmatrix} 0 & 2 \\ 1 & -1 \end{vmatrix} = 0 - 2 = -2$
$M_{13} = \begin{vmatrix} 0 & -1 \\ 1 & 4 \end{vmatrix} = 0 + 1 = 1$
Step 3
Apply Cofactor Signs
Multiply each minor by $(-1)^{i+j}$: signs alternate $+, -, +$ across the first row.
$A_{11} = +(-7) = -7$, $A_{12} = -(-2) = 2$, $A_{13} = +(1) = 1$
Step 4
Sum the Products
Multiply each element by its cofactor and add.
$|A| = 2(-7) + 3(2) + 1(1) = -14 + 6 + 1 = -7$
Memorize formulas
Cofactor Sign
The sign of a cofactor depends on its position in the checkerboard pattern.
Formula
$A_{ij} = (-1)^{i+j} M_{ij}$
2×2 Adjoint
For a 2×2 matrix: swap diagonal, negate off-diagonal.
Formula
$$\text{adj}\begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$$
Inverse Determinant
The determinant of an inverse equals the reciprocal of the original determinant.
Formula
$|A^{-1}| = \frac{1}{|A|}$
2x2 Determinant
Cross-multiply diagonals and subtract (Main - Secondary).
Formula
|A| = ad - bc
Matrix Inverse
The formula for the multiplicative inverse of any non-singular square matrix.
Formula
$A^{-1} = \frac{1}{|A|} \text{adj}(A)$
Product Transpose Rule
The 'Socks and Shoes' rule: the transpose of a product reverses the order.
Formula
$(AB)^T = B^T A^T$
Scalar Determinant Property
Scaling an entire $n \times n$ matrix raises the scalar factor to the power of $n$.
Formula
$|kA| = k^n |A|$
Product Determinant Rule
The determinant of a product is the product of individual determinants.
Formula
|AB| = |A||B|
Review
1
If $A$ is a $3 \times 3$ matrix with $|A| = -3$, what is $|A^2|$?
Ans
$9$
Review
Tap to reveal
Use the product rule for determinants
A matrix square means the matrix is multiplied by itself, so the determinant also multiplies with itself.
$$|A^2| = |A||A|$$
Substitute the given determinant
Now replace $|A|$ with $-3$ and evaluate.
$$|A^2| = (-3)^2 = 9$$
2
If $\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} = 7$, what is $\begin{vmatrix} d & e & f \\ a & b & c \\ g & h & i \end{vmatrix}$?
Ans
$-7$
Review
Tap to reveal
Apply the row-swap rule
Swapping any two rows changes the sign of the determinant and does nothing else.
Reverse the original value
The original determinant is 7, so after one swap it becomes negative.
$$\det = -7$$
3
If $A$ is a $3 \times 3$ non-singular matrix and $|\text{adj}(A)| = 9$, what is $|A|$?
Ans
$3$
Review
Tap to reveal
Use the adjugate determinant formula
For an $n \times n$ matrix, $|\operatorname{adj}(A)| = |A|^{n-1}$. With $n=3$, this becomes a square.
$$|\operatorname{adj}(A)| = |A|^2$$
Solve the resulting equation
Substitute the given value and take the square root to match the listed option.
$$9 = |A|^2 \Rightarrow |A| = 3$$
4
If $|A| = 3$ and $|B| = -2$ for two $n \times n$ matrices, what is $|A^{-1}B|$?
Ans
$-2/3$
Review
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Invert the determinant of A
For an invertible matrix, the determinant of the inverse is the reciprocal of the original determinant.
$$|A^{-1}| = \frac{1}{|A|} = \frac{1}{3}$$
Multiply the two determinants
Now use the product rule for determinants of matrix products.
$$|A^{-1}B| = \frac{1}{3} \cdot (-2) = -\frac{2}{3}$$
5
If $A$ is symmetric ($A^T = A$) and $B$ is skew-symmetric ($B^T = -B$), what is $(AB + BA)^T$?
Ans
$-(AB + BA)$
Review
Tap to reveal
Transpose each product separately
Use $(XY)^T = Y^T X^T$ and distribute the transpose across the sum.
$$ (AB+BA)^T = (AB)^T + (BA)^T $$
Substitute the symmetry conditions
Replace $A^T$ with $A$ and $B^T$ with $-B$, then simplify the order of factors.
$$ (AB+BA)^T = B^TA^T + A^TB^T = (-B)A + A(-B) = -(AB+BA) $$
6
What is the determinant of $\begin{bmatrix} 3 & 1 & 5 \\ 0 & -2 & 4 \\ 0 & 0 & 7 \end{bmatrix}$?
Ans
$-42$
Review
Tap to reveal
Use the triangular-matrix rule
For any upper or lower triangular matrix, the determinant is the product of the diagonal entries.
Multiply the diagonal entries
Only the diagonal terms matter here, so compute their product directly.
$$3 \cdot (-2) \cdot 7 = -42$$
7
For which value of $k$ is the matrix $\begin{bmatrix} 1 & 2 & 3 \\ 0 & k & 1 \\ 2 & 4 & 6 \end{bmatrix}$ singular?
Ans
All values of $k$
Review
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Check for proportional rows
Row 3 is exactly 2 times Row 1, so the matrix already has two dependent rows before using k at all.
$$R_3 = 2R_1$$
Conclude singularity
A matrix with proportional rows has determinant zero, so it is singular for every value of k.
$$\det(A) = 0 \text{ for all } k$$
8
If $a_{12} = 3$, $M_{12} = 5$, then the cofactor $A_{12}$ equals:
Ans
$-5$
Review
Tap to reveal
Apply the cofactor sign pattern
A cofactor is the minor multiplied by $(-1)^{i+j}$, so the position determines the sign.
$$A_{12} = (-1)^{1+2} M_{12}$$
Substitute the minor
Use the given minor value and evaluate the sign.
$$A_{12} = (-1)^3 \cdot 5 = -5$$
9
The inverse of $\begin{bmatrix} 3 & 1 \\ 5 & 2 \end{bmatrix}$ is:
Ans
$\begin{bmatrix} 2 & -1 \\ -5 & 3 \end{bmatrix}$
Review
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Find the determinant first
A 2x2 inverse exists only after confirming the determinant, computed as $ad-bc$.
$$|A| = 3 \cdot 2 - 1 \cdot 5 = 1$$
Form the inverse matrix
Swap the diagonal entries, change the signs of the off-diagonals, and divide by the determinant.
$$A^{-1} = \frac{1}{1}\begin{bmatrix} 2 & -1 \\ -5 & 3 \end{bmatrix} = \begin{bmatrix} 2 & -1 \\ -5 & 3 \end{bmatrix}$$
10
If column 2 of a $4 \times 4$ matrix is exactly twice column 1, what can be concluded about the determinant?
Ans
The determinant is zero
Review
Tap to reveal
Recognize dependent columns
Since one column is a scalar multiple of another, the columns are linearly dependent.
$$C_2 = 2C_1$$
Use the zero-determinant rule
Any matrix with proportional columns has determinant zero.
$$\det(A) = 0$$
11
If $A$ and $B$ are $2 \times 2$ matrices with $|A| = -1$ and $|B| = 3$, what is $|A^2 B^T|$?
Ans
$3$
Review
Tap to reveal
Apply determinant rules to each factor
Use the product rule, the power rule for A squared, and the transpose rule for B.
$$|A^2B^T| = |A|^2 \cdot |B|$$
Substitute the given values
Now evaluate the expression using the provided determinants.
$$(-1)^2 \cdot 3 = 3$$
Question 1 of 11
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1
If $A$ is a $3 \times 3$ matrix with $|A| = -3$, what is $|A^2|$?